Clustered Variants of Haj\'os' Conjecture

TL;DR

This paper proves that graphs excluding certain subdivisions can be colored with a bounded number of colors such that each monochromatic component is small, extending Hajós' conjecture to a clustered coloring context.

Contribution

It introduces new bounds on the clustered chromatic number for graphs excluding subdivisions and minors, generalizing Hajós' conjecture to a weaker, clustered coloring setting.

Findings

Graphs of bounded treewidth with no almost $( extless=1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering.

Graphs with no $H$-minor and no almost $( extless=1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering.

Graphs with no $H$-subdivision and no almost $( extless=1)$-subdivision of $K_{s+1}$ are $ ext{max}\{s+3d-5,2 ight\}$-colorable with bounded clustering.

Abstract

Haj\'os conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\Omega(s^2/\log s)$. We prove that $O(s)$ colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called clustered coloring). Our approach leads to more results. Say that a graph is an almost $(\leq 1)$-subdivision of a graph $H$ if it can be obtained from $H$ by subdividing edges, where at most one edge is subdivided more than once. Note that every graph with no $H$-subdivision does not contain an almost $(\leq 1)$-subdivision of $H$. We prove the following (where $s \geq 2$): (1) Graphs of bounded treewidth and with no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering. (2) For every graph $H$, graphs with no $H$-minor and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering. (3) For every graph $H$ of maximum degree at most $d$, graphs with no $H$-subdivision and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $\max\{s+3d-5,2\}$-colorable with bounded clustering. (4) For every graph $H$ of maximum degree $d$, graphs with no $K_{s,t}$ subgraph and no $H$-subdivision are $\max\{s+3d-4,2\}$-colorable with bounded clustering. (5) Graphs with no $K_{s+1}$-subdivision are $(4s-5)$-colorable with bounded clustering. The first result shows that the weakening of Haj\'{o}s' conjecture is true for graphs of bounded treewidth in a stronger sense; the final result is the first $O(s)$ bound on the clustered chromatic number of graphs with no $K_{s+1}$-subdivision.

Equations

$$N_G^{⩾s}(X)$$

$$N_G^{<s}(X)$$

$$|L(y)|=s+r-|N_G(y)\cap Y_1|⩾r+1$$

$$|N_G(v)\cap ((Y_1\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}}))\cup (S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})))|⩾d-1.$$

$$Z_1$$

$$Z_2$$

$$Z_3$$

$$|Z_1\cup Z_2\cup Z_3|$$

$$=|V(A_x\cap B_x)|-|V(A_x\cap B_x)-V(B_{\mathcal{C}})|$$

$$⩽d-1-|V(A_x\cap B_x)-V(B_{\mathcal{C}})|.$$

$$S^{\prime}:=\big{(}S\cap V(B_{x})-V(A_{x}\cap B_{x}\cap A_{\mathcal{C}}\cap B_{\mathcal{C}})\big{)}\cup\big{(}V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\big{)}\cup V(A_{x}\cap B_{x}\cap A_{\mathcal{C}}\cap B_{\mathcal{C}}).$$

$$|S^{\prime }|$$

$$⩽|S\cap V(B_x)|-|\{y\in X^{\prime }:\iota (y)\in Z_3\}|+|X|$$

$$=|S\cap V(B_x)|+|X-\{y\in X^{\prime }:\iota (y)\in Z_3\}|$$

$$=|\{y\in Y_1\cap V(A_{\mathcal{C}}):\iota (y)\in S\cap V(B_x)\}|+|X-X^{\prime }|+|X^{\prime }-\{y\in X^{\prime }:\iota (y)\in Z_3\}|.$$

$$\iota (y)\in N_G(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cap V(A_x)-V(A_x\cap B_x\cap A_{\mathcal{C}}\cap B_{\mathcal{C}})=N_G(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}\cap A_x)-V(B_x),$$

$$|S^{\prime }|⩽|\{y\in Y_1\cap V(A_{\mathcal{C}}):\iota (y)\in S\cap V(B_x)\}\cup (X-\{y\in X^{\prime }:\iota (y)\in Z_3\})|.$$

$$T^{\prime }:=T\cup (Y_1\cap V(B_{\mathcal{C}})-V(A_{\mathcal{C}}\cup B_x))\cup (Y_1\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-(V(B_x)\cup N_G[V(A_{{\mathcal{C}}^{\prime }}\cap B_{{\mathcal{C}}^{\prime }})])).$$

$$|S^{\prime }|+|T^{\prime }|+|Z|+1$$

$$⩽|S|+|T|+|Y_1\cap V(B_{\mathcal{C}})-V(A_{\mathcal{C}}\cup B_x)|+|Z|+1$$

$$⩽|Y_1\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})|+|\{y\in Y_1\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}):\iota (y)=y\}|$$

$$+|Y_1\cap (V(A_{{\mathcal{C}}^{\prime }})-V(B_{{\mathcal{C}}^{\prime }}))-V(A_{\mathcal{C}})|$$

$$⩽|Y_1\cap V(A_{{\mathcal{C}}^{\prime }})-V(B_{{\mathcal{C}}^{\prime }})|-|Y_1\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_x)|$$

$$+|\{y\in Y_1\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}):\iota (y)=y\}|$$

$$⩽|Y_1\cap V(A_{{\mathcal{C}}^{\prime }})-V(B_{{\mathcal{C}}^{\prime }})|+|\{y\in Y_1\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cap V(B_x):\iota (y)=y\}|$$

$$⩽|Y_1\cap V(A_{{\mathcal{C}}^{\prime }})-V(B_{{\mathcal{C}}^{\prime }})|+|\{y\in Y_1\cap S^{\prime }\cap V(A_{{\mathcal{C}}^{\prime }}\cap B_{{\mathcal{C}}^{\prime }}):{\iota }^{\prime }(y)=y\}|.$$

$$2d-4$$

$$=|((Y_1\cap V(A_{\mathcal{C}}))\cup (S^{\prime }-V(A_{{\mathcal{C}}^{\prime }}\cap B_{{\mathcal{C}}^{\prime }})))\cap N_G(v^{\prime })|$$

$$⩽|((Y_1\cap V(A_{\mathcal{C}}))\cup (S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})))\cap N_G(v^{\prime })|$$

$$<2d-4,$$

$$|N_G(v^{\prime })\cap ((Y_1\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}}))\cup (S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}}))|$$

$$⩾$$

$$=$$

$$⩾$$

$$=$$

$$G_B$$

$$Y_B$$

$$|Y_B|$$

$$⩽(|Y_1|-|Y_1\cap V(A_{{\mathcal{C}}^{\ast }})-V(B_{{\mathcal{C}}^{\ast }})|)+|S^{\ast }|-|Y_1\cap V(A_{{\mathcal{C}}^{\ast }}\cap B_{{\mathcal{C}}^{\ast }})|+|Z|+|T^{\ast }|$$

$$⩽|Y_1|-1$$

$$|N_G(v)\cap Y_A|$$

$$⩽|N_G(v)\cap V(A_{{\mathcal{C}}^{\ast }}\cap B_{{\mathcal{C}}^{\ast }})|+|N_G(v)\cap Z|+|N_G(v)\cap ((Y_1\cap V(A_{{\mathcal{C}}^{\ast }}))\cup (S^{\ast }-V(A_{{\mathcal{C}}^{\ast }}\cap B_{{\mathcal{C}}^{\ast }})))|$$

$$⩽(d-1)+(s-1)+(2d-5)$$

Full text

Our approach leads to more results, many of which only require a much weaker assumption that forbids an ‘almost $(\leqslant 1)$-subdivision’ (where at most one edge is subdivided more than once). This assumption is best possible, since no bound on the number of colors exists unless we allow at least one edge to be subdivided arbitrarily many times. We prove the following (where $s\geqslant 2$):

Graphs of bounded treewidth and with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering. 2. 2.

For every graph $H$, graphs with no $H$-minor and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering. 3. 3.

For every graph $H$ of maximum degree at most $d$, graphs with no $H$-subdivision and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ are $\max\{s+3d-5,2\}$-colorable with bounded clustering. 4. 4.

For every graph $H$ of maximum degree $d$, graphs with no $K_{s,t}$ subgraph and no $H$-subdivision are $\max\{s+3d-4,2\}$-colorable with bounded clustering. 5. 5.

Graphs with no $K_{s+1}$-subdivision are $(4s-5)$-colorable with bounded clustering.

The first result is tight and shows that the clustered analogue of Hajós’ conjecture is true for graphs of bounded treewidth. The second result implies an upper bound for the clustered version of Hadwiger’s conjecture that is only one color away from the known lower bound, and shows that the number of colors is independent of the forbidden minor. The final result is the first $O(s)$ bound on the clustered chromatic number of graphs with no $K_{s+1}$-subdivision.

1 Introduction

In the 1940s, Hajós conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is $s$-colorable; see [31, 30, 25]. Dirac [5] proved the conjecture for $s\leqslant 3$. It is open for $s\in\{4,5\}$, which would imply the Four Color Theorem. Catlin [3] presented counterexamples for all $s\geqslant 6$, and Erdős and Fajtlowicz [8] proved that the conjecture is false for almost all graphs. Indeed, there are graphs with no $K_{s+1}$-subdivision and with chromatic number $\Omega(s^{2}/\log s)$. The best upper bound on the number of colors is $O(s^{2})$, independently due to Bollobás and Thomason [2] and Komlós and Szemerédi [19]; see [12] for a related result. See [25, 31] for more explicit counterexamples and further discussion of connections to other areas of graph theory.

The purpose of this paper is to prove several positive results in the direction of weakenings of Hajós’ conjecture. Define a coloring of a graph $G$ to simply be a function that assigns one color to each vertex of $G$. For a coloring $c$ of a graph $G$, a monochromatic $c$-component of $G$ is a connected component of a subgraph of $G$ induced by all the vertices assigned the same color by $c$. When $c$ is clear, we simply write monochromatic component. A coloring has clustering $\eta$ if every monochromatic component has at most $\eta$ vertices. Our focus is on minimizing the number of colors, with small clustering as a secondary goal. The clustered chromatic number of a graph class $\mathcal{F}$ is the minimum integer $k$ for which there exists an integer $c$ such that every graph in $\mathcal{F}$ has a $k$-coloring with clustering $c$. There have been several recent papers on this topic [28, 15, 16, 1, 4, 17, 18, 20, 10, 13, 11, 6, 21, 14, 26]; see [33] for a survey.

Most of our results actually hold (in some sense) for more general classes of graphs than those with no $K_{s+1}$-subdivision, as we now explain. Say a graph $H^{\prime}$ is an almost $(\leqslant 1)$-subdivision of a graph $H$ if $H^{\prime}$ can be obtained from $H$ by subdividing edges, where at most one edge is subdivided more than once. Most of our results say that all graphs containing no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, plus some other properties, are $s$-colorable with bounded clustering.

The following is our first main result. It provides a Hajós-type result for clustered coloring of graphs with bounded treewidth.

Theorem 1**.**

For all $s,w\in\mathbb{N}$, there exists $\eta\in{\mathbb{N}}$ such that every graph with treewidth at most $w$ and with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ is $s$-choosable with clustering $\eta$.

The notion of $s$-choosable with bounded clustering is defined in Section 2.1. Note that every graph that is $s$-choosable with bounded clustering is also $s$-colorable with bounded clustering. This shows that the number of colors in Theorem 1 is best possible in the following strong sense: for all $s\in\mathbb{N}$ and $\eta\in\mathbb{N}$ there is a graph $G$ with treewidth at most $s-1$ (and thus with no subdivision of $K_{s+1}$), such that every $(s-1)$-coloring of $G$ has a monochromatic component with at least $\eta$ vertices; see [33]. In particular, at least $s$ colors are required even for this weakening of Hajós’ conjecture.

The assumption of bounded treewidth in Theorem 1 is equivalent to saying that the graph excludes a planar graph as a minor by Robertson and Seymour’s Grid Minor Theorem [29]. What if we exclude a general graph as a minor? Our next result answers this question (with one more color).

Theorem 2**.**

For every $s\in\mathbb{N}$ and every graph $H$, there exists $\eta\in{\mathbb{N}}$ such that every graph containing no $H$-minor and containing no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ is $(s+1)$-colorable with clustering $\eta$.

Theorem 2 (with $H=K_{s+1}$) has the following interesting corollary for graphs excluding a minor.

Corollary 3**.**

For every $s\in\mathbb{N}$ there exists $\eta\in{\mathbb{N}}$ such that every graph containing no $K_{s+1}$-minor is $(s+1)$-colorable with clustering $\eta$.

Kawarabayashi and Mohar [18] first proved that graphs containing no $K_{s+1}$-minor are $O(s)$-colorable with bounded clustering. The bound on the number of colors has since been steadily improved [32, 7, 21, 27, 15]. Prior to the present work, the best bound was $s+2$, which followed from a general result by the authors [23]. Corollary 3 improves this bound to $s+1$, although it should be noted that results from [23] are essential for the proof of Theorems 2 and 3. Dvořák and Norin [6] have announced that a forthcoming paper will prove that $s$ colors suffice (which is the clustered analogue of Hadwiger’s Conjecture, and would be best possible). Their result is incomparable with Theorem 2 and the aforementioned general result in [23].

Our next result relaxes the assumption that the graph contains no $H$-minor, and instead assumes that it contains no $H$-subdivision. The price paid is an increase in the number of colors, depending only on the maximum degree of $H$.

Theorem 4**.**

For every $s\in\mathbb{N}$ and every graph $H$ with maximum degree $d\in\mathbb{N}$, there exists $\eta\in\mathbb{N}$ such that every graph with no $H$-subdivision and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ is $\max\{s+3d-5,2\}$-colorable with clustering $\eta$.

The next theorem relaxes the assumption of no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, and instead assumes the graph contains no $K_{s,t}$-subgraph. Interestingly the number of colors does not depend on $t$. Note that $K_{s,t}$ contains a $K_{s+1}$-subdivision where every edge is subdivided at most once, when $t$ is sufficiently large.

Theorem 5**.**

For $s,t,d\in\mathbb{N}$ and every graph $H$ of maximum degree $d$, there exists $\eta\in{\mathbb{N}}$ such that every graph with no $K_{s,t}$-subgraph and no $H$-subdivision is $\max\{s+3d-4,2\}$-colorable with clustering $\eta$.

We remark that all of the above theorems forbid $(\leqslant 1)$-subdivisions of $K_{s+1}$ or subdivisions of $H$. That is, we forbid a subdivision of a graph where some edge is allowed to be subdivided arbitrarily many times. This condition is required since there are graphs of arbitrarily high girth and arbitrarily high chromatic number [9], which therefore require arbitrarily many colors for any fixed clustering value; this shows that excluding finitely many graphs as subgraphs cannot ensure any upper bound on the number of colors.

Our final theorem simply excludes a $K_{s+1}$-subdivision. This is the first $O(s)$ bound on the clustered chromatic number of the class of graphs excluding a $K_{s+1}$-subdivision.

Theorem 6**.**

For each $s\in\mathbb{N}$, there exists $\eta\in{\mathbb{N}}$ such that every graph containing no $K_{s+1}$-subdivision is $\max\{4s-5,1\}$-colorable with clustering $\eta$.

We now compare the above theorems with Hajós’ conjecture. First note that Theorems 1–4 are stronger than Hajós’ conjecture in the sense that they only exclude an almost $(\leqslant 1)$-subdivision of $K_{s+1}$, whereas Hajós’ conjecture excludes all subdivisions of $K_{s+1}$. Moreover, Theorem 1 also holds in the stronger setting of choosability. On the other hand, Theorems 1–6 are weaker than Hajós’ conjecture in the sense that they have bounded clustering rather than a proper coloring. However, such a weakening is unavoidable since Hajós’ conjecture is false. Indeed, the proof of the theorem of Erdős and Fajtlowicz [8] mentioned above shows that, for a suitable constant $c$, almost every graph on $cs^{2}$ vertices contains no subdivision of $K_{s+1}$ and has chromatic number $\Omega(s^{2}/\log s)$. Trivially, such a graph has treewidth at most $cs^{2}$ and contains no $K_{cs^{2}}$-minor. Thus the clustering function in all of the above theorems is at least $\Omega(s/\log s)$.

The paper is organized as follows. Section 2 introduces preliminary definitions and results from our companion papers [24, 23] that are used in the present paper. Section 3 introduces a structure theorem of the first author and Thomas [22] for graphs excluding a fixed subdivision, and uses it to prove Theorem 5. Building on this work, Section 4 proves the remaining theorems mentioned above.

2 Preliminaries

We use the following notation. Let $\mathbb{N}_{0}:=\{0,1,2,\dots\}$ and $\mathbb{N}:=\{1,2,\dots\}$. For $m,n\in\mathbb{N}_{0}$, let $[m,n]:=\{m,m+1,\dots,n\}$ and $[n]:=[1,n]$.

Let $G$ be a graph (allowing loops and parallel edges). For $v\in V(G)$, let $N_{G}(v):=\{w\in V(G):vw\in E(G)\}$ be the neighborhood of $v$, and let $N_{G}[v]:=N_{G}(v)\cup\{v\}$. For $X\subseteq V(G)$, let $N_{G}(X):=\bigcup_{v\in X}(N_{G}(v)-X)$ and $N_{G}[X]:=N_{G}(X)\cup X$. Denote the subgraph of $G$ induced by $X$ by $G[X]$.

For a graph $G$, a subset $X$ of $V(G)$, and an integer $s\geqslant 1$, let

[TABLE]

When the graph $G$ is clear from the context we write $N^{\geqslant s}(X)$ instead of $N_{G}^{\geqslant s}(X)$, and similarly for $N^{<s}(X)$.

Lemma 7** ([24, Lemma 12]).**

For all $s,t\in\mathbb{N}$, there exists a function $f_{s,t}:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$ such that for every graph $G$ with no $K_{s,t}$ subgraph, if $X\subseteq V(G)$ then $|N^{\geqslant s}(X)|\leqslant f_{s,t}(\lvert X\rvert)$.

Lemma 7 is sufficient to prove the theorems in this paper. But when $G$ excludes a fixed minor or subdivision of a fixed graph, the function $f_{s,t}$ in Lemma 7 can be made linear; see [24]. This improves the clustering function in all our results, although to simplify the presentation, we choose not to explicitly evaluate our clustering functions.

A tree-decomposition of a graph $G$ is a pair $(T,\mathcal{X}=(X_{x}:x\in V(T)))$, where $T$ is a tree and for each node $x\in V(T)$, $X_{x}$ is a subset of $V(G)$ called a bag, such that for each vertex $v\in V(G)$, the set $\{x\in V(T):v\in X_{x}\}$ induces a non-empty (connected) subtree of $T$, and for each edge $vw\in E(G)$ there is a node $x\in V(T)$ such that $\{v,w\}\subseteq X_{x}$. The width of a tree-decomposition $(T,\mathcal{X})$ is $\max\{|X_{x}|-1:x\in V(T)\}$. The treewidth of a graph $G$ is the minimum width of a tree-decomposition of $G$.

Let $H$ be a graph. An $H$-minor of a graph $G$ is a map $\alpha$ with domain $V(H)\cup E(H)$ such that:

• •

For every $h\in V(H)$, $\alpha(h)$ is a nonempty connected subgraph of $G$.

• •

If $h_{1}$ and $h_{2}$ are different vertices of $H$, then $\alpha(h_{1})$ and $\alpha(h_{2})$ are disjoint.

• •

For each edge $e$ of $H$ with endpoints $h_{1},h_{2}$, $\alpha(e)$ is an edge of $G$ with one end in $\alpha(h_{1})$ and one end in $\alpha(h_{2})$; furthermore, if $h_{1}=h_{2}$, then $\alpha(e)\in E(G)-E(\alpha(h_{1}))$.

• •

If $e_{1},e_{2}$ are two different edges of $H$, then $\alpha(e_{1})\neq\alpha(e_{2})$.

2.1 List Coloring

For our purposes, a color is an element of $\mathbb{Z}$. Let $G$ be a graph. A list-assignment of $G$ is a function $L$ with domain containing $V(G)$, such that $L(v)$ is a non-empty set of colors for each vertex $v\in V(G)$. For a list-assignment $L$ of $V(G)$, an $L$-coloring of $G$ is a coloring $c$ of $G$ such that $c(v)\in L(v)$ for every $v\in V(G)$. An $L$-coloring has clustering $\eta$ if every monochromatic component has at most $\eta$ vertices. A list-assignment $L$ of $G$ is an $\ell$-list-assignment if $|L(v)|\geqslant\ell$ for every vertex $v\in V(G)$. A graph $G$ is $\ell$-choosable with clustering $\eta$ if $G$ is $L$-colorable with clustering $\eta$ for every $\ell$-list-assignment $L$ of $G$.

For a graph $G$, a subset $Y_{1}\subseteq V(G)$, and $s,r\in\mathbb{N}$, a list-assignment $L$ of $G$ is an $(s,r,Y_{1})$-list-assignment if:

• (L1)

$\lvert L(v)\rvert\in[s+r]$ for every $v\in V(G)$.

• (L2)

$Y_{1}=\{v\in V(G):\lvert L(v)\rvert=1\}$.

• (L3)

For every $y\in N^{<s}(Y_{1})$,

[TABLE]

and $L(y)\cap L(u)=\emptyset$ for every $u\in N_{G}(y)\cap Y_{1}$.

• (L4)

For every $v\in V(G)-N_{G}[Y_{1}]$, we have $\lvert L(v)\rvert=s+r$.

• (L5)

For every $v\in V(G)-Y_{1}$, we have $\lvert L(v)\rvert\geqslant r+1$.

We say that an $(s,r,Y_{1})$-list-assignment $L$ of $G$ is restricted if:

• (L1’)

$L(v)\subseteq[s+r]$ for every $v\in V(G)$.

Note that a restricted $(s,2,Y_{1})$-list-assignment is called a $(s,Y_{1},0,0)$-list-assignment in our companion paper [23].

For a list-assignment $L$ of a graph $G$ with $Y_{1}=\{v\in V(G):\lvert L(v)\rvert=1\}$, for $\eta\in\mathbb{N}$ and a nondecreasing function $g:\mathbb{N}\to\mathbb{N}$, an $L$-coloring $c$ of $G$ is $(\eta,g)$-bounded if:

• •

the union of the monochromatic components intersecting $Y_{1}$ contains at most $\lvert Y_{1}\rvert^{2}g(\lvert Y_{1}\rvert)$ vertices, and

• •

every monochromatic component contains at most $\eta^{2}g(\eta)$ vertices.

2.2 Companion Results

Our companion paper proves the following results for graphs with no $K_{s,t}$ subgraph. The first assumes bounded treewidth, the second assumes an excluded minor.

Theorem 8** ([23, Theorem 17]).**

For all $s,t,w\in\mathbb{N}$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that if $G$ is a graph of treewidth at most $w$ and with no $K_{s,t}$ subgraph, $Y_{1}$ is a subset of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and $L$ is an $(s,1,Y_{1})$-list-assignment of $G$, then there exists an $(\eta,g)$-bounded $L$-coloring of $G$.

Theorem 9** ([23, Theorem 24]).**

For all $s,t\in\mathbb{N}$ and for every graph $H$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that if $G$ is a graph with no $K_{s,t}$ subgraph and no $H$-minor, $Y_{1}$ is a subset of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and $L$ is a restricted $(s,2,Y_{1})$-list-assignment of $G$, then there exists an $(\eta,g)$-bounded $L$-coloring.

2.3 Progress

The concept of “progress” from the proofs of the above two theorems are re-used in the present paper. Let $s,r\in\mathbb{N}$ and $L$ be an $(s,r,Y_{1})$-list-assignment of a graph $G$. For $W\subseteq V(G)$, a $W$-progress of $L$ is a list-assignment $L^{\prime}$ of $G$ defined as follows:

• •

Let $Y_{1}^{\prime}:=Y_{1}\cup W$.

• •

For every $y\in Y_{1}$, let $L^{\prime}(y):=L(y)$.

• •

For every $y\in Y_{1}^{\prime}-Y_{1}$, let $L^{\prime}(y)$ be a 1-element subset of $L(y)$ (which exists by (L2)–(L5)).

• •

For each $v\in N^{<s}(Y_{1}^{\prime})$, let $L^{\prime}(v)$ be a subset of $L(v)-\{L^{\prime}(w):w\in N_{G}(v)\cap(W-Y_{1})\}$ of size $\lvert L(v)\rvert-\lvert N_{G}(v)\cap(W-Y_{1})\rvert$.

• •

For every $v\in V(G)-(Y_{1}^{\prime}\cup N^{<s}(Y_{1}^{\prime}))$, let $L^{\prime}(v):=L(v)$.

Lemma 10** ([23, Lemma 12 with $F=\emptyset$]).**

Let $s,r\in\mathbb{N}$ and $L$ be an $(s,r,Y_{1})$-list-assignment of a graph $G$. Let $W\subseteq V(G)$. Then every $W$-progress $L^{\prime}$ of $L$ is an $(s,r,Y_{1}\cup W)$-list-assignment of $G$, and $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$.

Lemma 11** ([23, Lemma 13 with $F=\emptyset$]).**

For all $s,t,k\in\mathbb{N}$, there exist a number $\eta>k$ and a nondecreasing function $g$ with domain ${\mathbb{N}_{0}}$ and with $g(0)\geqslant\eta$ such that if $G$ is a graph with no $K_{s,t}$ subgraph, $r\in\mathbb{N}$, $Y_{1}$ is a subset of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and $L$ is an $(s,r,Y_{1})$-list-assignment of $G$, then at least one of the following holds:

There exists an $(\eta,g)$-bounded $L$-coloring of $G$. 2. 2.

$\lvert Y_{1}\rvert>k$. 3. 3.

For every color $\ell$, there exist a subset $Y_{1}^{\prime}$ of $V(G)$ with $\eta\geqslant\lvert Y_{1}^{\prime}\rvert>\lvert Y_{1}\rvert$ and an $(s,r,Y_{1}^{\prime})$-list-assignment $L^{\prime}$ of $G$ with $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$, such that:

1. (a)

there does not exist an $(\eta,g)$-bounded $L^{\prime}$-coloring $c^{\prime}$ of $G$, 2. (b)

for every $L^{\prime}$-coloring of $G$, every monochromatic component intersecting $Y_{1}$ is contained in $G[Y_{1}^{\prime}]$, and 3. (c)

for every $y\in Y_{1}^{\prime}$ with $\ell\in L^{\prime}(y)$, we have $\{v\in N_{G}(y)-Y_{1}^{\prime}:\ell\in L^{\prime}(v)\}=\emptyset$. 4. 4.

$Y_{1}\neq\emptyset$, $N_{G}(Y_{1})=\emptyset$, and there does not exist an $(\eta,g)$-bounded $L|_{G-Y_{1}}$-coloring of $G-Y_{1}$.

2.4 Separations and Tangles

A separation of a graph $G$ is an ordered pair $(A,B)$ of edge-disjoint subgraphs of $G$ with $A\cup B=G$. The order of $(A,B)$ is $\lvert V(A\cap B)\rvert$. A tangle ${\mathcal{T}}$ in a graph $G$ of order $\theta$ is a set of separations of $G$ of order less than $\theta$ such that:

• (T1)

For every separation $(A,B)$ of $G$ of order less than $\theta$, either $(A,B)\in{\mathcal{T}}$ or $(B,A)\in{\mathcal{T}}$.

• (T2)

If $(A_{i},B_{i})\in{\mathcal{T}}$ for $i\in[3]$, then $A_{1}\cup A_{2}\cup A_{3}\neq G$.

• (T3)

If $(A,B)\in{\mathcal{T}}$, then $V(A)\neq V(G)$.

Lemma 12** ([23, Lemma 16 with $F=\emptyset$]).**

For all $s,t,\theta,\eta,r\in\mathbb{N}$ with $\eta\geqslant 9\theta+1$, for every nondecreasing function $g$ with domain ${\mathbb{N}_{0}}$, if $G$ is a graph with no $K_{s,t}$ subgraph, $Y_{1}$ is a subset of $V(G)$ with $9\theta+1\leqslant\lvert Y_{1}\rvert\leqslant\eta$, and $L$ is an $(s,r,Y_{1})$-list-assignment of $G$, then at least one of the following holds:

There exists an $(\eta,g)$-bounded $L$-coloring of $G$. 2. 2.

There exist an induced subgraph $G^{\prime}$ of $G$ with $\lvert V(G^{\prime})\rvert<\lvert V(G)\rvert$, a subset $Y_{1}^{\prime}$ of $V(G^{\prime})$ with $\lvert Y_{1}^{\prime}\rvert\leqslant\eta$ and an $(s,r,Y_{1}^{\prime})$-list-assignment $L^{\prime}$ of $G^{\prime}$ such that:

1. (a)

$L^{\prime}(v)\subseteq L(v)$* for every $v\in V(G^{\prime})$.* 2. (b)

There does not exist an $(\eta,g)$-bounded $L^{\prime}$-coloring of $G^{\prime}$. 3. 3.

${\mathcal{T}}:=\{(A,B):\lvert V(A\cap B)\rvert<\theta,\lvert V(A)\cap Y_{1}\rvert\leqslant 3\theta\}$* is a tangle of order $\theta$ in $G$.*

A tangle ${\mathcal{T}}$ in $G$ controls an $H$-minor $\alpha$ if there does not exist $(A,B)\in{\mathcal{T}}$ of order less than $\lvert V(H)\rvert$ such that $V(\alpha(h))\subseteq V(A)$ for some $h\in V(H)$.

Lemma 13** ([23, Lemma 23 with $\ell=r=0$]).**

For all $s,t,t^{\prime}\in\mathbb{N}$, there exist $\theta^{*}\in\mathbb{N}$ and nondecreasing functions $g^{*},\eta^{*}$ with domain ${\mathbb{N}_{0}}$ such that if $G$ is a graph with no $K_{s,t}$ subgraph, $\theta\in\mathbb{N}$ with $\theta\geqslant\theta^{*}$, $\eta\in\mathbb{N}$ with $\eta\geqslant\eta^{*}(\theta)$, $Y_{1}\subseteq V(G)$ with $3\theta<\lvert Y_{1}\rvert\leqslant\eta$, $L$ is a restricted $(s,2,Y_{1})$-list-assignment of $G$, $g$ is a nondecreasing function with domain ${\mathbb{N}_{0}}$ with $g\geqslant g^{*}$, and ${\mathcal{T}}:=\{(A,B):\lvert V(A\cap B)\rvert<\theta,\lvert V(A)\cap Y_{1}\rvert\leqslant 3\theta\}$ is a tangle in $G$ of order $\theta$ that does not control a $K_{t^{\prime}}$-minor, then either:

there exists an $(\eta,g)$-bounded $L$-coloring of $G$, or 2. 2.

there exist $(A^{*},B^{*})\in{\mathcal{T}}$, a set $Y_{A^{*}}$ with $\lvert Y_{A^{*}}\rvert\leqslant\eta^{*}(\theta)$ and $Y_{1}\cap V(A^{*})\subseteq Y_{A^{*}}\subseteq V(A^{*})$, and a restricted $(s,2,Y_{A^{*}})$-list-assignment $L_{A^{*}}$ of $G[V(A^{*})]$ such that there exists no $(\eta,g)$-bounded $L_{A^{*}}$-coloring of $G[V(A^{*})]$.

3 Excluding Subdivisions

The following theorem is a special case of a theorem by the first author and Thomas [22].

Theorem 14** ([22, Theorem 6.8]).**

For any integers $d,h$ and graph $H$ on $h$ vertices with maximum degree at most $d$, there exist integers $\theta,\xi$ such that if $G$ is a graph containing no $H$-subdivision, and if ${\mathcal{T}}$ is a tangle in $G$ of order at least $\theta$ controlling a $K_{\lfloor\frac{3}{2}dh\rfloor}$-minor, then there exists $Z\subseteq V(G)$ with $\lvert Z\rvert\leqslant\xi$ such that for every vertex $v\in V(G)-Z$, there exists $(A,B)\in{\mathcal{T}}-Z$ of order less than $d$ such that $v\in V(A)-V(B)$.

The next two lemmas imply Theorem 5, since if $s,d\in{\mathbb{N}}$ and $3d+s<7$, then $d=1$.

Lemma 15**.**

If $H$ is a graph of maximum degree at most 1, then every graph with no $H$-subdivision is 2-colorable with clustering $\max\{2\lvert V(H)\rvert-2,1\}$.

Proof.

Since $H$ is of maximum degree at most one, $G$ has no $H$-subdivision implies that $G$ does not contain a matching of size $\lvert V(H)\rvert$, and hence $G$ contains a vertex-cover $S$ of size at most $2\lvert V(H)\rvert-2$. By coloring every vertex in $S$ with 1 and coloring every vertex in $V(G)-S$ with 2, we obtain a 2-coloring of $G$ with clustering $\max\{\lvert S\rvert,1\}\leqslant\max\{2\lvert V(H)\rvert-2,1\}$. ∎

Lemma 16**.**

For any $s,t,d\in\mathbb{N}$ and graph $H$ of maximum degree $d$ with $3d+s\geqslant 7$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that if $G$ is a graph with no $K_{s,t}$ subgraph and no $H$-subdivision, $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$ and $L$ is a restricted $(s^{\prime},2,Y_{1})$-list-assignment of $G$, then there exists an $(\eta,g)$-bounded $L$-coloring, where $s^{\prime}=3d+s-6$.

Proof.

Define the following:

• •

Let $f$ be the function $f_{s,t}$ mentioned in Lemma 7.

• •

Let $\theta_{0}$ be the number $\theta^{*}$ and $g_{0},\eta_{0}$ be the functions $g^{*},\eta^{*}$, respectively, mentioned in Lemma 13 by taking $s=s^{\prime},t=t$ and $t^{\prime}=\lfloor\frac{3}{2}d\lvert V(H)\rvert\rfloor$.

• •

Let $\theta_{1}$ and $\xi$ be the numbers $\theta$ and $\xi$ mentioned in Theorem 14, respectively, by taking $d=d$, $h=\lvert V(H)\rvert$ and $H=H$.

• •

Let $a_{0}:=f(\xi)d^{2}+\xi+1$, and let $a_{i}:=da_{i-1}+1$ for $i\in\mathbb{N}$.

• •

Let $\theta:=\theta_{0}+\theta_{1}+(d-1)a_{(d-1)a_{0}}$.

• •

Let $\eta_{1}$ be the number $\eta$ and let $g_{1}$ be the function $g$ mentioned in Lemma 11 by taking $s=s^{\prime}$, $t=t$ and $k=9\theta$. Note that $g(0)\geqslant\eta_{1}>9\theta$ by Lemma 11.

• •

Let $\eta:=\eta_{0}(\theta)+\eta_{1}+(d-1)a_{(d-1)a_{0}}$.

• •

Let $g:\mathbb{N}\to\mathbb{N}$ be the function defined by $g(0):=g_{0}(0)+g_{1}(0)$ and $g(x+1):=g_{0}(x+1)+g_{1}(x+1)+\sum_{i=0}^{x}i^{2}g(i)$ for $x\in\mathbb{N}$.

Let $G$ be a graph with no $K_{s,t}$ subgraph and with no subdivision of $H$, let $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and let $L$ be a restricted $(s^{\prime},2,Y_{1})$-list-assignment of $G$. Suppose to the contrary that there exists no $(\eta,g)$-bounded $L$-coloring of $G$. We further assume that $\lvert V(G)\rvert$ is minimum, and subject to this, $\lvert Y_{1}\rvert$ is maximum.

Claim 16.1**.**

$Y_{1}\neq\emptyset$* and $N_{G}(Y_{1})\neq\emptyset$.*

Proof.

First suppose that $Y_{1}=\emptyset$. Let $v$ be a vertex of $G$, and let $L^{\prime}$ be a $\{v\}$-progress of $L$. Let $Y_{1}^{\prime}=\{v\}$. By Lemma 10, $L^{\prime}$ is an $(s^{\prime},2,Y_{1}^{\prime})$-list-assignment of $G$. Since $\lvert Y_{1}^{\prime}\rvert\leqslant\eta$, the maximality of $\lvert Y_{1}\rvert$ implies that there exists an $(\eta,g)$-bounded $L^{\prime}$-coloring $c^{\prime}$ of $G$. But $c^{\prime}$ is an $(\eta,g)$-bounded $L$-coloring $c$ of $G$, a contradiction.

So $Y_{1}\neq\emptyset$. Suppose that $N_{G}(Y_{1})=\emptyset$. Let $G^{\prime}:=G-Y_{1}$. Then $L|_{G^{\prime}}$ is an $(s^{\prime},2,\emptyset)$-list-assignment of $G-Y_{1}$. By the minimality of $\lvert V(G)\rvert$, there exists an $(\eta,g)$-bounded $L|_{G^{\prime}}$-coloring $c$ of $G^{\prime}$. Color each vertex $y$ in $Y_{1}$ with the unique element in $L(y)$. Since $\lvert Y_{1}\rvert\leqslant\lvert Y_{1}\rvert^{2}g(\lvert Y_{1}\rvert)$, we obtain an $(\eta,g)$-bounded $L$-coloring of $G$, a contradiction. ∎

Claim 16.2**.**

$\lvert Y_{1}\rvert\geqslant 9\theta+1$.

Proof.

Suppose $\lvert Y_{1}\rvert\leqslant 9\theta$. So $\lvert Y_{1}\rvert<\eta_{1}$. Since $G$ has no $K_{s,t}$ subgraph, $G$ has no $K_{s^{\prime},t}$ subgraph. Applying Lemmas 11 and 16.1, either there exists an $(\eta_{1},g_{1})$-bounded $L$-coloring of $G$, or there exist $Y_{1}^{\prime}\subseteq V(G)$ with $\eta_{1}\geqslant\lvert Y_{1}^{\prime}\rvert>\lvert Y_{1}\rvert$ and an $(s^{\prime},2,Y_{1}^{\prime})$-list-assignment $L^{\prime}$ of $G$ with $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$ such that for every $L^{\prime}$-coloring of $G$, every monochromatic component intersecting $Y_{1}$ is contained in $G[Y_{1}^{\prime}]$. Since $\eta_{1}\leqslant\eta$ and $g_{1}\leqslant g$, every $(\eta_{1},g_{1})$-bounded $L$-coloring of $G$ is an $(\eta,g)$-bounded $L$-coloring of $G$, so the former does not hold. Hence there exist $Y_{1}^{\prime}\subseteq V(G)$ with $\eta_{1}\geqslant\lvert Y_{1}^{\prime}\rvert>\lvert Y_{1}\rvert$ and a restricted $(s^{\prime},2,Y_{1}^{\prime})$-list-assignment $L^{\prime}$ of $G$ with $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$ such that for every $L^{\prime}$-coloring of $G$, every monochromatic component intersecting $Y_{1}$ is contained in $G[Y_{1}^{\prime}]$. Since $\lvert Y_{1}^{\prime}\rvert\leqslant\eta_{1}\leqslant\eta$, the maximality of $\lvert Y_{1}\rvert$ implies that there exists an $(\eta,g)$-bounded $L^{\prime}$-coloring $c^{\prime}$ of $G$. So every monochromatic component respect to $c^{\prime}$ contains at most $\eta^{2}g(\eta)$ vertices. Since $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$, $c^{\prime}$ is also an $L$-coloring of $G$. Every monochromatic $c^{\prime}$-component intersecting $Y_{1}$ is contained in $G[Y_{1}^{\prime}]$ and hence contains at most $\lvert Y_{1}^{\prime}\rvert\leqslant\eta_{1}\leqslant g_{1}(0)\leqslant g(0)\leqslant\lvert Y_{1}\rvert^{2}g(\lvert Y_{1}\rvert)$ vertices. So $c^{\prime}$ is an $(\eta,g)$-bounded $L$-coloring of $G$, a contradiction. ∎

Let ${\mathcal{T}}$ be the set of separations $(A,B)$ of $G$ such that $\lvert V(A\cap B)\rvert<\theta$ and $\lvert V(A)\cap Y_{1}\rvert\leqslant 3\theta$.

Claim 16.3**.**

${\mathcal{T}}$* is a tangle in $G$ of order $\theta$.*

Proof.

Suppose that ${\mathcal{T}}$ is not a tangle in $G$ of order $\theta$. Note that $G$ has no $K_{s^{\prime},t}$ subgraph and $L$ is an $(s^{\prime},2,Y_{1})$-list-assignment of $G$ with $\eta\geqslant\lvert Y_{1}\rvert\geqslant 9\theta+1$ by 16.2. Applying Lemma 12 by taking $s=s^{\prime}$, $t=t$, $\theta=\theta$, $\eta=\eta$, $r=2$ and $g=g$, there exists an induced subgraph $G^{\prime}$ of $G$ with $\lvert V(G^{\prime})\rvert<\lvert V(G)\rvert$, a subset $Y_{1}^{\prime}\subseteq V(G^{\prime})$ with $\lvert Y_{1}^{\prime}\rvert\leqslant\eta$, and an $(s^{\prime},2,Y_{1}^{\prime})$-list-assignment $L^{\prime}$ of $G^{\prime}$ with $L^{\prime}(v)\subseteq L(v)$ for every $v\in V(G)$ such that there exists no $(\eta,g)$-bounded $L^{\prime}$-coloring of $G^{\prime}$. This contradicts the minimality of $\lvert V(G)\rvert$. ∎

Claim 16.4**.**

${\mathcal{T}}$* controls a $K_{\lfloor\frac{3}{2}d\lvert V(H)\rvert\rfloor}$-minor.*

Proof.

Suppose to the contrary that ${\mathcal{T}}$ does not control a $K_{\lfloor\frac{3}{2}d\lvert V(H)\rvert\rfloor}$-minor. Note that $\theta\geqslant\theta_{0}$, $\eta\geqslant\eta_{0}(\theta)$ and $g\geqslant g_{0}$. Apply Lemma 13 with $s=s^{\prime}$, $t=t$ and $t^{\prime}=\lfloor\frac{3}{2}d\lvert V(H)\rvert\rfloor$. Since there does not exist an $(\eta,g)$-bounded $L$-coloring of $G$, we know there exist $(A^{*},B^{*})\in{\mathcal{T}}$, a set $Y_{A^{*}}$ with $\lvert Y_{A^{*}}\rvert\leqslant\eta_{0}(\theta)\leqslant\eta$ and $Y_{1}\cap V(A^{*})\subseteq Y_{A^{*}}\subseteq V(A^{*})$, and a restricted $(s^{\prime},2,Y_{A^{*}})$-list-assignment $L_{A^{*}}$ of $G[V(A^{*})]$ such that there exists no $(\eta,g)$-bounded $L_{A^{*}}$-coloring of $G[V(A^{*})]$. But $\lvert V(A^{*})\rvert<\lvert V(G)\rvert$ since $\lvert V(A^{*})\cap Y_{1}\rvert<3\theta<\lvert Y_{1}\rvert$. This contradicts the minimality of $\lvert V(G)\rvert$. ∎

Since $G$ contains no subdivision of $H$, by Theorem 14 and 16.4, there exists $Z\subseteq V(G)$ with $\lvert Z\rvert\leqslant\xi$ such that for every $v\in V(G)-Z$, there exists $(A_{v},B_{v})\in{\mathcal{T}}-Z$ of order at most $d-1$ such that $v\in V(A_{v})-V(B_{v})$.

We may assume that for every $v\in V(G)-Z$,

• (i)

$(A_{v},B_{v})\in{\mathcal{T}}-Z$ has order at most $d-1$ and $v\in V(A_{v})-V(B_{v})$,

• (ii)

subject to (i), $A_{v}-V(A_{v}\cap B_{v})$ is connected,

• (iii)

subject to (i) and (ii), every vertex in $V(A_{v}\cap B_{v})$ is adjacent to some vertex in $V(A_{v})-V(B_{v})$,

• (iv)

subject to (i)–(iii), $V(A_{v})$ is maximal,

• (v)

subject to (i)–(iv), $\lvert V(A_{v}\cap B_{v})\rvert$ is minimal, and

• (vi)

subject to (i)–(v), $A_{v}$ is maximal.

Note that for every $v\in V(G)-Z$, $A_{v}$ is connected and for every two vertices $x,y\in V(A_{v})$, there exists a path in $A_{v}$ from $x$ to $y$ internally disjoint from $V(A_{v}\cap B_{v})$ since $A_{v}-V(A_{v}\cap B_{v})$ is connected and every vertex in $V(A_{v}\cap B_{v})$ is adjacent to some vertex in $V(A_{v})-V(B_{v})$

For any subset ${\mathcal{C}}\subseteq{\mathcal{T}}-Z$, let $(A_{\mathcal{C}},B_{\mathcal{C}})$ be the separation $(\bigcup_{(A,B)\in{\mathcal{C}}}A,\bigcap_{(A,B)\in{\mathcal{C}}}B)$. Note that $V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq\bigcup_{(A,B)\in{\mathcal{C}}}V(A\cap B)$, so $\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert\leqslant\lvert{\mathcal{C}}\rvert(d-1)$.

Claim 16.5**.**

Let ${\mathcal{C}}=\{(A_{w},B_{w}):w\in W\}$ for some $W\subseteq V(G)-Z$. If $x$ is a vertex in $V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, then $V(A_{x}\cap B_{x})-V(B_{w})\neq\emptyset$ for some $w\in V(G)-Z$ with $(A_{w},B_{w})\in{\mathcal{C}}$.

Proof.

Since $x\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, there exists $w\in W\subseteq V(G)-Z$ such that $(A_{w},B_{w})\in{\mathcal{C}}$ and $x\in V(A_{w}\cap B_{w})$. Suppose to the contrary that $V(A_{x}\cap B_{x})\subseteq V(B_{w})$.

First suppose that there exists $v\in V(A_{w})-(V(B_{w})\cup V(A_{x}))$. Since $A_{w}-V(B_{w})$ is connected by (ii) and every vertex in $V(A_{w}\cap B_{w})$ is adjacent to a vertex in $V(A_{w})-V(B_{w})$ by (iii), there exists a path $P$ in $G[(V(A_{w})-V(B_{w}))\cup\{x\}]$ from $x$ to $v$. Since $x\in V(A_{x})-V(B_{x})$ and $v\in V(G)-(Z\cup V(A_{x}))$, $P-x$ intersects $V(A_{x}\cap B_{x})\subseteq V(B_{w})$. But $V(P-x)\subseteq V(A_{w})-V(B_{w})$, a contradiction. So $V(A_{w})-V(B_{w})\subseteq V(A_{x})$.

Suppose that there exists a vertex $u\in V(A_{w}\cap B_{w})-V(A_{x})$. Since $u\in V(A_{w}\cap B_{w})$, there exists $u^{\prime}\in N_{G}(u)\cap V(A_{w})-V(B_{w})$ by (iii). So $u^{\prime}\in N_{G}(u)\cap V(A_{w})-V(B_{w})\subseteq N_{G}(u)\cap V(A_{x})$. Since $u\not\in V(A_{x})$, $u^{\prime}\in V(A_{x}\cap B_{x})\cap V(A_{w})-V(B_{w})$, contradicting the assumption $V(A_{x}\cap B_{x})\subseteq V(B_{w})$. Hence $V(A_{w}\cap B_{w})\subseteq V(A_{x})$.

Therefore, $V(A_{w})\subseteq V(A_{x})$. By (v), every vertex in $V(A_{x}\cap B_{x})$ is adjacent to some vertex in $V(B_{x})-V(A_{x})$. So if $V(A_{x})=V(A_{w})$, then $V(A_{x}\cap B_{x})\subseteq V(A_{w}\cap B_{w})$, and since $(A_{w},B_{w})$ satisfies (v), $V(B_{w})=V(B_{x})$. Hence if $V(A_{x})=V(A_{w})$, then $(A_{x},B_{x})=(A_{w},B_{w})$ by (vi). Since $x\in V(A_{x})-V(B_{x})$ and $x\in V(A_{w}\cap B_{w})$, $(A_{x},B_{x})\neq(A_{w},B_{w})$. So $V(A_{w})\subset V(A_{x})$. Since $(A_{w},B_{w})$ satisfies (iv), $w\in V(B_{x})$. Since $V(A_{x}\cap B_{x})\subseteq V(B_{w})$ and $w\not\in V(B_{w})$, $w\in V(B_{x})-V(A_{x})$. So $V(A_{w})\not\subseteq V(A_{x})$, a contradiction. ∎

Let $Z^{\prime}:=\{v\in V(G)-(Y_{1}\cup Z):\lvert N_{G}(v)\cap Z\rvert\geqslant s\}$. Note that $\lvert Z^{\prime}\rvert\leqslant f(\lvert Z\rvert)\leqslant f(\xi)$ by Lemma 7.

We say that a triple $({\mathcal{C}},S,T)$ is useful if the following hold:

• (U1)

There exists $W\subseteq V(G)-Z$ such that ${\mathcal{C}}=\{(A_{v},B_{v}):v\in W\}$.

• (U2)

$N_{G}[N_{G}[Z^{\prime}]]\cap V(B_{\mathcal{C}})=\emptyset$.

• (U3)

$S$ is a subset of $N_{G}[V(A_{\mathcal{C}}\cap B_{\mathcal{C}})]\cap V(A_{\mathcal{C}})$ and $T$ is a subset of $Y_{1}\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$ such that there exists a bijection $\iota$ from a subset of $Y_{1}\cap V(A_{\mathcal{C}})$ to $S$ such that:

• –

$\lvert S\rvert+\lvert T\rvert+\lvert Z\rvert+1\leqslant\lvert Y_{1}\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert+\lvert\{y\in Y_{1}\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}):\iota(y)=y\}\rvert$, and

• –

for every vertex $y$ in the domain of $\iota$,

if $y\in V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$ and there exists a vertex $v\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$, then $\iota(y)\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, and

• *

if $y\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, then $\iota(y)=y$.

• (U4)

$T$ is disjoint from $Z^{\prime}$ and the domain of $\iota$.

Claim 16.6**.**

There exists a collection ${\mathcal{C}}$ of members of ${\mathcal{T}}-Z$ with $\lvert{\mathcal{C}}\rvert\leqslant\lvert Z^{\prime}\rvert d^{2}+\lvert Z\rvert+1$ such that $({\mathcal{C}},\emptyset,\emptyset)$ is useful.

Proof.

For every $u\in V(G)-Z$, let ${\mathcal{C}}_{u}:=\{(A_{u},B_{u}),(A_{v},B_{v}):v\in N_{G}(u)\cap V(B_{u})\}$. Note that $\lvert N_{G}(u)\cap V(B_{u})\rvert\leqslant\lvert V(A_{u}\cap B_{u})\rvert\leqslant d-1$ since $u\in V(A_{u})-V(B_{u})$. So $\lvert{\mathcal{C}}_{u}\rvert\leqslant d$. Note that $N_{G}[\{u\}]\cap V(B_{{\mathcal{C}}_{u}})=\emptyset$.

For every $u\in V(G)-Z$, let ${\mathcal{C}}_{u}^{\prime}:={\mathcal{C}}_{u}\cup\{(A_{v},B_{v}):v\in N_{G}(N_{G}[\{u\}])\cap V(B_{{\mathcal{C}}_{u}})\}$. Note that $\lvert N_{G}(N_{G}[\{u\}])\cap V(B_{{\mathcal{C}}_{u}})\rvert\leqslant\lvert V(A_{{\mathcal{C}}_{u}}\cap B_{{\mathcal{C}}_{u}})\rvert\leqslant(d-1)\lvert{\mathcal{C}}_{u}\rvert\leqslant(d-1)d$. So $\lvert{\mathcal{C}}_{u}^{\prime}\rvert\leqslant\lvert{\mathcal{C}}_{u}\rvert+(d-1)d\leqslant d^{2}$. Note that $N_{G}[N_{G}[\{u\}]]\cap V(B_{{\mathcal{C}}_{u}^{\prime}})=\emptyset$.

Let ${\mathcal{C}}^{\prime}:=\bigcup_{z\in Z^{\prime}}{\mathcal{C}}_{z}^{\prime}$. Then $N_{G}[N_{G}[Z^{\prime}]]\cap V(B_{{\mathcal{C}}^{\prime}})=\emptyset$. And $\lvert{\mathcal{C}}^{\prime}\rvert\leqslant\lvert Z^{\prime}\rvert d^{2}$. Since $\lvert Y_{1}-Z\rvert\geqslant\lvert Y_{1}\rvert-\lvert Z\rvert\geqslant 9\theta-\xi\geqslant 8\theta>\lvert Z\rvert$, there exists a subset $Y$ of $Y_{1}-Z$ with $\lvert Y\rvert=\lvert Z\rvert+1$. Let ${\mathcal{C}}:={\mathcal{C}}^{\prime}\cup\{(A_{y},B_{y}):y\in Y\}$. Clearly $({\mathcal{C}},\emptyset,\emptyset)$ satisfies (U1) and (U4). Since $B_{{\mathcal{C}}^{\prime}}\supseteq B_{\mathcal{C}}$, $({\mathcal{C}},\emptyset,\emptyset)$ satisfies (U2). Since $Y\subseteq V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$, $\lvert Z\rvert+1=\lvert Y\rvert\leqslant\lvert Y_{1}\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert$, so $({\mathcal{C}},\emptyset,\emptyset)$ satisfies (U3). Note that $\lvert C\rvert\leqslant\lvert{\mathcal{C}}^{\prime}\rvert+\lvert Y\rvert\leqslant\lvert Z^{\prime}\rvert d^{2}+\lvert Z\rvert+1$. ∎

For a useful triple $({\mathcal{C}},S,T)$, a vertex $v$ of $V(G)-Z$ is:

• •

$({\mathcal{C}},S,T)$-dangerous if $v\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$ and there exists $v^{\prime}\in N_{G}(v)\cap V(A_{\mathcal{C}})-(V(B_{\mathcal{C}})\cup S)$ such that either:

• –

$v^{\prime}\not\in Y_{1}$ and $\lvert((Y_{1}\cap V(A_{\mathcal{C}}))\cup(S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$, or

• –

$v^{\prime}\in Y_{1}-T$,

• •

$({\mathcal{C}},S,T)$-heavy if $v\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$ and

[TABLE]

Claim 16.7**.**

Let $({\mathcal{C}},S,T)$ be a useful triple and let $x\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ be a $({\mathcal{C}},S,T)$-heavy vertex. Then there exists a useful triple $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ with ${\mathcal{C}}^{\prime}={\mathcal{C}}\cup\{(A_{x},B_{x})\}$, such that:

• •

$V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$,

• •

the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-heavy vertices is strictly contained in the set of $({\mathcal{C}},S,T)$-heavy vertices, and

• •

the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is a subset of the set of $({\mathcal{C}},S,T)$-dangerous vertices.

Proof.

Let ${\mathcal{C}}^{\prime}:={\mathcal{C}}\cup\{(A_{x},B_{x})\}$. Let $X:=N_{G}(x)\cap(Y_{1}\cup S)\cap V(A_{x}\cap A_{\mathcal{C}})-V(B_{x}\cup B_{\mathcal{C}})$. Since $x\in V(A_{x})-V(B_{x})$, $N_{G}(x)\subseteq V(A_{x})$. So $\lvert X\rvert\geqslant\lvert N_{G}(x)\cap(Y_{1}\cup S)\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert-\lvert V(A_{x}\cap B_{x}\cap A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert\geqslant d-1-\lvert V(A_{x}\cap B_{x}\cap A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert$ since $x$ is $({\mathcal{C}},S,T)$-heavy. That is, $\lvert V(A_{x}\cap B_{x})-V(B_{\mathcal{C}})\rvert=\lvert V(A_{x}\cap B_{x}\cap A_{\mathcal{C}})-V(B_{\mathcal{C}})\rvert\geqslant d-1-\lvert X\rvert$. Since $x$ is $({\mathcal{C}},S,T)$-heavy, $x\not\in S$. Let $\iota$ be a bijection mentioned in (U3) witnessing that $({\mathcal{C}},S,T)$ is useful. Let $X^{\prime}$ be the intersection of $X$ and the domain of $\iota$.

For each $y\in X^{\prime}$, since $x\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$, $\iota(y)\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ by (U3). Since $X^{\prime}\subseteq X\subseteq V(A_{x}\cap A_{\mathcal{C}})-V(B_{x}\cup B_{\mathcal{C}})$, for each $y\in X^{\prime}$, $N_{G}(y)\subseteq V(A_{x})$, so $\iota(y)\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq V(A_{x})\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq(V(A_{{\mathcal{C}}^{\prime}})-V(B_{{\mathcal{C}}^{\prime}}))\cup V(A_{{\mathcal{C}}}\cap B_{{\mathcal{C}}}\cap A_{x}\cap B_{x})$. Let

[TABLE]

So $\{Z_{1},Z_{2},Z_{3}\}$ is a partition of $V(A_{x}\cap B_{x}\cap B_{\mathcal{C}})$, and hence

[TABLE]

Recall that $\lvert V(A_{x}\cap B_{x})-V(B_{\mathcal{C}})\rvert\geqslant d-1-\lvert X\rvert$. So $\lvert Z_{1}\cup Z_{2}\cup Z_{3}\rvert\leqslant(d-1)-(d-1-\lvert X\rvert)=\lvert X\rvert$.

Let

[TABLE]

Note that $(V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cup V(A_{x}\cap B_{x}\cap A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq V(A_{x}\cap B_{x}\cap B_{\mathcal{C}})=Z_{1}\cup Z_{2}\cup Z_{3}$. So

[TABLE]

Recall that for every $y\in X^{\prime}$, $\iota(y)\in N_{G}(y)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cap V(A_{x})$. So if $y\in X^{\prime}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\}$, then

[TABLE]

so $\iota(y)\not\in S\cap V(B_{x})$. That is, $\{y\in Y_{1}\cap V(A_{\mathcal{C}}):\iota(y)\in S\cap V(B_{x})\}$ and $X^{\prime}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\}$ are disjoint. Note that $X-X^{\prime}$ is disjoint from the domain of $\iota$. So $\{y\in Y_{1}\cap V(A_{\mathcal{C}}):\iota(y)\in S\cap V(B_{x})\}$, $X-X^{\prime}$ and $X^{\prime}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\}$ are pairwise disjoint sets. Therefore,

[TABLE]

Since $X\subseteq Y_{1}\cup S$ and $X\cap V(B_{x})=\emptyset$, for every $x\in X-\{y\in X^{\prime}:\iota(y)\in Z_{3}\}$, if $x\not\in X\cap Y_{1}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\}$, then $x\in X\cap S-Y_{1}$ and $\iota(y)=x$ for some $y\in Y_{1}\cap V(A_{\mathcal{C}})$ such that $\iota(y)\not\in S\cap V(B_{x})$. In addition, if $y$ is a vertex in $Y_{1}\cap V(A_{\mathcal{C}})$ such that $\iota(y)\in X\cap S-Y_{1}$, then $\iota(y)\not\in S\cap V(B_{x})$.

Since $\lvert S^{\prime}\rvert\leqslant\lvert\{y\in Y_{1}\cap V(A_{\mathcal{C}}):\iota(y)\in S\cap V(B_{x})\}\cup(X-\{y\in X^{\prime}:\iota(y)\in Z_{3}\})\rvert$, there exists an injection $\iota^{\prime}$ such that

• •

$\iota^{\prime}(y)=\iota(y)$ if $y$ is in the domain of $\iota$ and $\iota(y)\in S\cap V(B_{x})$,

• •

for each $v\in S^{\prime}-(S\cap V(B_{x}))$, there exists exactly one element $y\in(X\cap Y_{1}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\})\cup\{y\in Y_{1}\cap V(A_{\mathcal{C}}):\iota(y)\in X\cap S-Y_{1}\}$ such that $\iota^{\prime}(y)=v$, and

• •

if $\iota(y_{1})=\iota^{\prime}(y_{2})$ for some $y_{1},y_{2}$, then $y_{1}=y_{2}$.

Recall that $\iota(y)\not\in S\cap V(B_{x})$ for every $y\in(X\cap Y_{1}-\{y\in X^{\prime}:\iota(y)\in Z_{3}\})\cup\{y\in Y_{1}:\iota(y)\in X\cap S-Y_{1}\}$. Then $\iota^{\prime}$ is a bijection from a subset of $Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}})$ to $S^{\prime}$. We further modify $\iota^{\prime}$ and $S^{\prime}$ by applying the following operations for some vertex $y\in V(A_{{\mathcal{C}}^{\prime}})-V(B_{{\mathcal{C}}^{\prime}})$ in the domain of $\iota^{\prime}$ with $\iota^{\prime}(y)\not\in N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$ and $N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}\neq\emptyset$, and then repeating until no such vertex $y$ exists:

• •

add a vertex $v\in N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}$ into $S^{\prime}$,

• •

delete $\iota^{\prime}(y)$ from $S^{\prime}$, and

• •

redefine $\iota^{\prime}(y)$ to be $v$.”

Now, further modify $\iota^{\prime}$ and $S^{\prime}$ by applying the following operations for some vertex $z\in S^{\prime}-N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]$, and repeating until no such vertex $z$ exists:

• •

remove $z$ from $S^{\prime}$, and

• •

if $y$ is the element in the domain of $\iota^{\prime}$ with $\iota^{\prime}(y)=z$, then remove $y$ from the domain of $\iota^{\prime}$.

Notice that for each vertex $z$ removed from $S^{\prime}$ in the above procedure, $z\in S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ and $N_{G}(z)\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_{x})$. Note that $\iota^{\prime}$ remains a bijection from a subset of $Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}})$ to $S^{\prime}$.

Observe that for every $y$ in the domain of $\iota^{\prime}$ with $y\in V(A_{\mathcal{C}}^{\prime})-V(B_{\mathcal{C}}^{\prime})$ and $N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}\neq\emptyset$, $\iota^{\prime}(y)\in N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$ due to the above modification. In addition, if $y$ is in the domain of $\iota^{\prime}$ and $y\in V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$, then $y\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ and $y$ is in the domain of $\iota$ such that $\iota(y)=\iota^{\prime}(y)$, so $\iota^{\prime}(y)=\iota(y)=y$.

Let $T^{\prime}$ be the set obtained from $T$ by deleting the domain of $\iota^{\prime}$. So $T^{\prime}$ is disjoint from the domain of $\iota^{\prime}$. Since $T$ is disjoint from $Z^{\prime}$, $T^{\prime}$ is disjoint from $Z^{\prime}$. So $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ satisfies (U4). In addition, $\lvert S^{\prime}\rvert-\lvert S\rvert$ is at most the number of vertices in $X$ and in the domain of $\iota^{\prime}$ but not in the domain of $\iota$. So $\lvert S^{\prime}\rvert-\lvert S\rvert\leqslant\lvert T\rvert-\lvert T^{\prime}\rvert$. Hence $\lvert S^{\prime}\rvert+\lvert T^{\prime}\rvert\leqslant\lvert S\rvert+\lvert T\rvert$. Since $\{y\in Y_{1}\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}):\iota(y)=y\}-\{y\in Y_{1}\cap S^{\prime}\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}}):\iota^{\prime}(y)=y\}\subseteq(V(A_{{\mathcal{C}}^{\prime}})-V(B_{{\mathcal{C}}^{\prime}}))-(V(A_{\mathcal{C}})-V(B_{\mathcal{C}}))$, $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ satisfies (U3) and is useful.

It is easy to see that $V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-S$. Note that each vertex $v\in V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}$ belongs to $V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(A_{x})$, so $N_{G}(v)\cap V(A_{{\mathcal{C}}^{\prime}})-V(B_{{\mathcal{C}}^{\prime}})=N_{G}(v)\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$. Furthermore, $S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})\subseteq S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$. Hence every $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-heavy vertex is $({\mathcal{C}},S,T)$-heavy. Since $x$ is $({\mathcal{C}},S,T)$-heavy but not $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-heavy, the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-heavy vertices is strictly contained in the set of $({\mathcal{C}},S,T)$-heavy vertices.

Let $v$ be a $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertex and let $v^{\prime}$ be a vertex in $N_{G}(v)\cap V(A_{{\mathcal{C}}^{\prime}})-(V(B_{{\mathcal{C}}^{\prime}})\cap S^{\prime})$ witnessing the definition of being dangerous. Since $v\not\in S^{\prime}$, $v\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(A_{x})$, so $v^{\prime}\in N_{G}[V(A_{\mathcal{C}}\cap B_{\mathcal{C}})]\cap V(A_{\mathcal{C}})\cap V(B_{x})$. So $v^{\prime}\in V(B_{x})-(V(B_{\mathcal{C}})\cup S^{\prime})$. Since $v\in V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$, $v^{\prime}\in N_{G}(v)\cap N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]$. Since $v^{\prime}\not\in S^{\prime}$ and $v^{\prime}\not\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$ and $v^{\prime}\in V(B_{x})$ and $N_{G}(v^{\prime})\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(A_{x})\neq\emptyset$, we know $v^{\prime}\not\in S$ by the procedure of modifying $S$. So $v^{\prime}\in(N_{G}(v)\cap V(A_{\mathcal{C}})-(V(B_{\mathcal{C}})\cup S))\cap V(B_{x})$. Note that $T-T^{\prime}\subseteq V(A_{x})-V(B_{x})$. So if $v^{\prime}\in Y_{1}-T^{\prime}$, then $v^{\prime}\in Y_{1}-T$ and $v$ is $({\mathcal{C}},S,T)$-dangerous. Furthermore, $Y_{1}\cap V(A_{\mathcal{C}})\cap N_{G}(v^{\prime})=Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}})\cap N_{G}(v^{\prime})$ and $S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})\subseteq S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, so $v$ is $({\mathcal{C}},S,T)$-dangerous. Therefore, the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is a subset of the set of $({\mathcal{C}},S,T)$-dangerous vertices. This proves the claim. ∎

Claim 16.8**.**

Let $({\mathcal{C}},S,T)$ be a useful triple. Then there exists a set $S^{\prime}$ with $S\cup(Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}))\subseteq S^{\prime}\subseteq N_{G}[V(A_{\mathcal{C}}\cap B_{\mathcal{C}})]\cap V(A_{\mathcal{C}})$ such that $({\mathcal{C}},S^{\prime},T)$ is a useful triple and:

• •

If $\iota^{\prime}$ is the bijection witnessing that $({\mathcal{C}},S^{\prime},T)$ satisfies (U3), then for every $y\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, the unique element of the domain of $\iota^{\prime}$ mapped to $y$ by $\iota^{\prime}$ is $y$.

• •

The set of $({\mathcal{C}},S^{\prime},T)$-dangerous vertices is contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices.

• •

The set of $({\mathcal{C}},S^{\prime},T)$-heavy vertices is contained in the set of $({\mathcal{C}},S,T)$-heavy vertices.

Proof.

Let $\iota$ be a function mentioned in (U3) witnessing that $({\mathcal{C}},S,T)$ is a useful triple. We may assume that $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq S$, since if some vertex $y\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ does not belong to $S$, then $y$ is not in the domain of $\iota$, and we can define $\iota(y)=y$ without violating (U3) and (U4) such that the set of dangerous vertices and the set of heavy vertices remain the same.

Since $\iota$ is a bijection, we write the element mapped to $y$ by $\iota$ as $\iota^{(-1)}(y)$. Modify $\iota$ and $S$ by applying the following operations to some vertex $y\in Y_{1}\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$ with $\iota^{(-1)}(y)\neq y$, and repeat until no such $y$ exists:

• •

remove $\iota^{(-1)}(y)$ from the domain of $\iota$,

• •

define $\iota(y):=y$,

Then define $S^{\prime}$ and $\iota^{\prime}$ to be the modified $S$ and $\iota$, respectively. Clearly, $({\mathcal{C}},S^{\prime},T)$ satisfies (U3), $S\subseteq S^{\prime}\subseteq N_{G}[V(A_{\mathcal{C}}\cap B_{\mathcal{C}})]\cap V(A_{\mathcal{C}})$, and $\iota^{\prime}(y)=y$ for every $y\in Y_{1}\cap S^{\prime}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$. Since we assume that $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq S$, we have $S\cup(Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}}))\subseteq S^{\prime}\subseteq N_{G}[V(A_{\mathcal{C}}\cap B_{\mathcal{C}})]\cap V(A_{\mathcal{C}})$ and $\iota^{\prime}(y)=y$ for every $y\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$. Since $T\subseteq V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$, $({\mathcal{C}},S^{\prime},T)$ satisfies (U4). Since $S^{\prime}-S\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, the set of $({\mathcal{C}},S^{\prime},T)$-dangerous vertices is contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices, and the set of $({\mathcal{C}},S^{\prime},T)$-heavy vertices is contained in the set of $({\mathcal{C}},S,T)$-heavy vertices. ∎

Claim 16.9**.**

Let $({\mathcal{C}},S,T)$ be a useful triple, and let $x$ be a $({\mathcal{C}},S,T)$-dangerous vertex. If there exists no $({\mathcal{C}},S,T)$-heavy vertex, then there exists a useful triple $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ with ${\mathcal{C}}^{\prime}={\mathcal{C}}\cup\{(A_{x},B_{x})\}$ such that the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is strictly contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices.

Proof.

By 16.8, we may assume that $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq S$ and the function $\iota$ mentioned in (U3) witnessing that $({\mathcal{C}},S,T)$ is useful satisfies $\iota(y)=y$ for every $y\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$. Let ${\mathcal{C}}^{\prime}:={\mathcal{C}}\cup\{(A_{x},B_{x})\}$.

We first assume that $\lvert Y_{1}\cap V(A_{x}\cap B_{\mathcal{C}})\rvert\geqslant d-2$. So $\lvert Y_{1}\cap V(A_{x}\cap B_{\mathcal{C}})\rvert\geqslant\lvert V(A_{x}\cap B_{x})\cap V(B_{\mathcal{C}})\rvert$ by 16.5. Hence there exists a function $\iota^{\prime}$ whose domain is a subset of $Y_{1}\cap V(A_{\mathcal{C}}\cup A_{x})$ such that:

• •

$\iota^{\prime}(y)=\iota(y)$ for every $y\in Y_{1}\cap V(A_{\mathcal{C}})-V(A_{x}\cap B_{\mathcal{C}})$ belonging to the domain of $\iota$ with $\iota(y)\in V(A_{\mathcal{C}})\cap N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]-V(A_{x})$, and

• •

for each vertex $v$ in $V(A_{x}\cap B_{x})\cap V(B_{\mathcal{C}})$, there exists exactly one element $y\in Y_{1}\cap V(A_{x}\cap B_{\mathcal{C}})$ such that $\iota^{\prime}(y)=v$ and if $v\in Y_{1}$, then $y=v$.

Let $S^{\prime}:=(S\cap N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]-V(A_{x}))\cup V(A_{x}\cap B_{x}\cap B_{\mathcal{C}})$. So $\iota^{\prime}$ is a bijection from a subset of $Y_{1}\cap V(A_{\mathcal{C}}\cup A_{x})$ to $S^{\prime}$. Note that every vertex in $S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$ is contained in $S-(V(A_{x})\cup V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}}))$, so it is adjacent to some vertex in $V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})$.

Let $T^{\prime}:=T$. Since $\iota$ satisfies (U3) and $\iota(y)=y$ for every $y\in Y_{1}\cap S\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, we know that $\iota^{\prime}$ satisfies (U3). Since $T^{\prime}=T\subseteq V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$, $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ satisfies (U4). So $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ is a useful triple.

Let $v$ be a $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertex. So $v\in V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S^{\prime}\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(A_{x})$. Let $v^{\prime}$ be a vertex witnessing that $v$ is $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous. So $v^{\prime}\in V(B_{x})-(V(B_{\mathcal{C}})\cup S^{\prime})$ and $N_{G}(v^{\prime})\subseteq V(A_{\mathcal{C}})$. Since $v\in V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-V(A_{x})$, $v^{\prime}\in N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]$, so $v^{\prime}\not\in S$. Since $S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})\subseteq S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, if $v^{\prime}\not\in Y_{1}$ and $\lvert((Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}}))\cup(S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$, then $v^{\prime}\not\in Y_{1}$ and $\lvert((Y_{1}\cap V(A_{\mathcal{C}}))\cup(S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$, so $v$ is $({\mathcal{C}},S,T)$-dangerous. Since $T^{\prime}=T$, if $v^{\prime}\in Y_{1}-T^{\prime}$, then $v^{\prime}\in Y_{1}-T$ and $v$ is $({\mathcal{C}},S,T)$-dangerous. So every $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertex is $({\mathcal{C}},S,T)$-dangerous. Since $x$ is $({\mathcal{C}},S,T)$-dangerous but not $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous, the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is strictly contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices. So the claim holds.

Hence we may assume that $\lvert Y_{1}\cap V(A_{x}\cap B_{\mathcal{C}})\rvert\leqslant d-3$.

Modify $S$ and define $\iota^{\prime}$ to be the function obtained from $\iota$ by applying the following operations to a vertex $y$ in the domain of $\iota$ with $\iota(y)\not\in N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]\cap V(A_{{\mathcal{C}}^{\prime}})$, and repeating until no such $y$ exists:

• •

if $y\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_{x})$ or $V(A_{x}\cap B_{x})\cap V(B_{\mathcal{C}})-S=\emptyset$, then remove $y$ from the domain of $\iota$ and remove $\iota(y)$ from $S$,

• •

if $y\not\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_{x})$ and $N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S=\emptyset$ and $V(A_{x}\cap B_{x})\cap V(B_{\mathcal{C}})-S\neq\emptyset$, then redefine $\iota(y)$ to be an element in $V(A_{x}\cap B_{x})\cap V(B_{\mathcal{C}})-S$ and add this element into $S$,

• •

otherwise remove $\iota(y)$ from $S$, redefine $\iota(y)$ to be an element in $N_{G}(y)\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})-S$ and add this element into $S$.

Let $S^{\prime}$ be the modified $S$, and let

[TABLE]

Clearly, $T^{\prime}$ is disjoint from the domain of $\iota^{\prime}$. By (U2), $Z^{\prime}\cap V(B_{\mathcal{C}})=\emptyset$. So $T^{\prime}$ is disjoint from $Z^{\prime}$ as $T$ is disjoint from $Z^{\prime}$. So $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ satisfies (U4).

Since $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq S$, we know $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-(V(B_{x})\cup N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})])\subseteq S$. For every $y\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-(V(B_{x})\cup N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})])$, since $\iota(y)=y\not\in N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]\cap V(A_{{\mathcal{C}}^{\prime}})$ and $y\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_{x})$, $y\in S-S^{\prime}$.

So $\lvert S\rvert\geqslant\lvert S^{\prime}\rvert+\lvert Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-(V(B_{x})\cup N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})])\rvert$. Hence $\lvert S^{\prime}\rvert+\lvert T^{\prime}\rvert\leqslant\lvert S\rvert+\lvert T\rvert+\lvert Y_{1}\cap V(B_{\mathcal{C}})-V(A_{\mathcal{C}}\cup B_{x})\rvert$. Since $Y_{1}\cap V(B_{\mathcal{C}})-V(A_{\mathcal{C}}\cup B_{x})\subseteq Y_{1}\cap(V(A_{{\mathcal{C}}^{\prime}})-V(B_{{\mathcal{C}}^{\prime}}))-V(A_{\mathcal{C}})$ and $({\mathcal{C}},S,T)$ satisfies (U3), we know

[TABLE]

Hence $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ satisfies (U3). Therefore $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ is useful.

Suppose that the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is not strictly contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices. Since $x$ is $({\mathcal{C}},S,T)$-dangerous but not $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous, there exists a vertex $v$ that is $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous but not $({\mathcal{C}},S,T)$-dangerous. So there exists a vertex $v^{\prime}\in N_{G}(v)\cap V(A_{{\mathcal{C}}^{\prime}})-(V(B_{{\mathcal{C}}^{\prime}})\cup S^{\prime})$ such that either $v^{\prime}\in Y_{1}-T^{\prime}$, or $v^{\prime}\not\in Y_{1}$ and $\lvert((Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}}))\cup(S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}}))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$. Since $v^{\prime}\in N_{G}[V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})]\cap V(A_{{\mathcal{C}}^{\prime}})$, if $v^{\prime}$ belongs to $S$ at beginning, then $v^{\prime}$ is not removed from $S$ during the process of modifying $S$, so $v^{\prime}\in S^{\prime}$, a contradiction. So $v^{\prime}\not\in S$.

Suppose that $v^{\prime}\in V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$. So $v\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cap V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})\subseteq V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\cap V(B_{x})$. Hence if $v$ belongs to $S$ at beginning, then $v$ is not removed from $S$ during the process of modifying $S$, so $v\in S^{\prime}$. Since $v$ is $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous, $v\not\in S^{\prime}$, so $v\not\in S$. Since $v$ is not $({\mathcal{C}},S,T)$-dangerous, $v^{\prime}\not\in Y_{1}-T$, and either $v^{\prime}\in Y_{1}$ or $\lvert((Y_{1}\cap V(A_{{\mathcal{C}}}))\cup(S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})))\cap N_{G}(v^{\prime})\rvert<2d-4$. Since $T^{\prime}\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})=T\cap V(A_{\mathcal{C}})-V(B_{\mathcal{C}})$, $v^{\prime}\not\in Y_{1}-T^{\prime}$. Since $v$ is $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous, $v^{\prime}\not\in Y_{1}$ and $\lvert((Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}}))\cup(S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$. Since $N_{G}(v^{\prime})\subseteq V(A_{\mathcal{C}})$ and $S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})\subseteq S-V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$,

[TABLE]

a contradiction.

Therefore, $v^{\prime}\in V(B_{\mathcal{C}})$. So $v^{\prime}\in(V(A_{x})-V(B_{x}))\cap V(B_{\mathcal{C}})$ and hence $N_{G}(v^{\prime})\subseteq V(A_{x})$. Since $Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\subseteq S$, if $v^{\prime}\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, then $v^{\prime}\in S$, a contradiction. So $v^{\prime}\not\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$. Since $Y_{1}\cap V(B_{\mathcal{C}})-V(A_{\mathcal{C}}\cup B_{x})\subseteq T^{\prime}$, if $v^{\prime}\in Y_{1}-T^{\prime}$, then $v^{\prime}\in Y_{1}\cap V(A_{\mathcal{C}}\cap B_{\mathcal{C}})$, a contradiction. So $v^{\prime}\not\in Y_{1}-T^{\prime}$. Since $v$ is $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous, $\lvert((Y_{1}\cap V(A_{{\mathcal{C}}^{\prime}}))\cup(S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}})))\cap N_{G}(v^{\prime})\rvert\geqslant 2d-4$. Since $\lvert Y_{1}\cap V(A_{x}\cap B_{\mathcal{C}})\rvert\leqslant d-3$ and $(S^{\prime}-V(A_{{\mathcal{C}}^{\prime}}\cap B_{{\mathcal{C}}^{\prime}}))-(V(A_{\mathcal{C}})\cup V(B_{x}))=\emptyset$ and $N_{G}(v^{\prime})\subseteq V(A_{x})$,

[TABLE]

In particular, $v^{\prime}\in V(A_{\mathcal{C}}\cap B_{\mathcal{C}})-V(B_{x})$. Since there exists no $({\mathcal{C}},S,T)$-heavy vertex, $v^{\prime}$ is not a $({\mathcal{C}},S,T)$-heavy vertex. So $v^{\prime}\in S$, a contradiction. This proves the claim. ∎

Claim 16.10**.**

If $({\mathcal{C}},S,T)$ is a useful triple such that there exists a $({\mathcal{C}},S,T)$-dangerous vertex, then there exists a useful triple $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ with ${\mathcal{C}}\subseteq{\mathcal{C}}^{\prime}$ and $\lvert{\mathcal{C}}^{\prime}\rvert\leqslant\lvert{\mathcal{C}}\rvert+\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert+1$ such that the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is strictly contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices.

Proof.

Note that there are at most $\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert$ $({\mathcal{C}},S,T)$-heavy vertices. By repeatedly applying 16.7 at most $\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert$ times, there exists a useful triple $({\mathcal{C}}_{1},S_{1},T_{1})$ with ${\mathcal{C}}\subseteq{\mathcal{C}}_{1}$ and $\lvert{\mathcal{C}}_{1}\rvert\leqslant\lvert{\mathcal{C}}\rvert+\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert$ such that there exists no $({\mathcal{C}}_{1},S_{1},T_{1})$-heavy vertices, and the set of $({\mathcal{C}}_{1},S_{1},T_{1})$-dangerous vertices is contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices. By 16.9 applied to ${\mathcal{C}}_{1}$, there exists a useful triple $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$ with ${\mathcal{C}}_{1}\subseteq{\mathcal{C}}^{\prime}$ and $\lvert{\mathcal{C}}^{\prime}\rvert=\lvert{\mathcal{C}}_{1}\rvert+1\leqslant\lvert{\mathcal{C}}\rvert+\lvert V(A_{\mathcal{C}}\cap B_{\mathcal{C}})\rvert+1$ such that the set of $({\mathcal{C}}^{\prime},S^{\prime},T^{\prime})$-dangerous vertices is strictly contained in the set of $({\mathcal{C}}_{1},S_{1},T_{1})$-dangerous vertices and hence is strictly contained in the set of $({\mathcal{C}},S,T)$-dangerous vertices. This proves the claim. ∎

Claim 16.11**.**

There exists a useful triple $({\mathcal{C}}^{*},S^{*},T^{*})$ with $\lvert{\mathcal{C}}^{*}\rvert\leqslant a_{(d-1)a_{0}}$ such that there exists no $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex.

Proof.

By 16.6, there exists a useful triple $({\mathcal{C}}_{0},\emptyset,\emptyset)$ with $\lvert{\mathcal{C}}_{0}\rvert\leqslant\lvert Z^{\prime}\rvert d^{2}+\lvert Z\rvert+1$. Let $S_{0}=\emptyset$ and $T_{0}=\emptyset$. So $({\mathcal{C}}_{0},S_{0},T_{0})$ is a useful triple with $\lvert{\mathcal{C}}_{0}\rvert\leqslant f(\xi)d^{2}+\xi+1=a_{0}$. For $i\geqslant 1$, if there exists a $({\mathcal{C}}_{i-1},S_{i-1},T_{i-1})$-dangerous vertex, then by 16.10, there exists a useful triple $({\mathcal{C}}_{i},S_{i},T_{i})$ such that $\lvert{\mathcal{C}}_{i}\rvert\leqslant\lvert{\mathcal{C}}_{i-1}\rvert+\lvert V(A_{{\mathcal{C}}_{i-1}}\cap B_{{\mathcal{C}}_{i-1}})\rvert+1$ and the set of $({\mathcal{C}}_{i},S_{i},T_{i})$-dangerous vertices is strictly contained in the set of $({\mathcal{C}}_{i-1},S_{i-1},T_{i-1})$-dangerous vertices. So $\lvert{\mathcal{C}}_{i}\rvert\leqslant a_{i-1}+(d-1)a_{i-1}+1\leqslant a_{i}$ for each $i\geqslant 1$ by induction on $i$. Since there are at most $\lvert V(A_{{\mathcal{C}}_{0}}\cap B_{{\mathcal{C}}_{0}})\rvert\leqslant\lvert{\mathcal{C}}_{0}\rvert(d-1)\leqslant(d-1)a_{0}$ $({\mathcal{C}}_{0},S_{0},T_{0})$-dangerous vertices. Hence there exists $i^{*}$ with $0\leqslant i^{*}\leqslant(d-1)a_{0}$ such that $({\mathcal{C}}_{i^{*}},S_{i^{*}},T_{i^{*}})$ is a useful triple with no $({\mathcal{C}}_{i^{*}},S_{i^{*}},T_{i^{*}})$-dangerous vertex. Note that $\lvert{\mathcal{C}}_{i^{*}}\rvert\leqslant a_{i^{*}}\leqslant a_{(d-1)a_{0}}$. ∎

Let $\iota^{*}$ be the function mentioned in (U3) witnessing that $({\mathcal{C}}^{*},S^{*},T^{*})$ is useful. By 16.8, we may assume that $Y_{1}\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\subseteq S^{*}$ such that $\iota^{*}(y)=y$ for every $y\in Y_{1}\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}*})$.

Define the following:

[TABLE]

Claim 16.12**.**

For every vertex $v\in V(G_{B})-Y_{B}$, $N_{G}(v)\cap Y_{1}\subseteq N_{G_{B}}(v)\cap Y_{B}$.

Proof.

Suppose to the contrary that there exist $v\in V(G_{B})-Y_{B}$ and $y\in N_{G}(v)\cap Y_{1}-(N_{G_{B}}(v)\cap Y_{B})$. Since $y\in Y_{1}-Y_{B}$, $y\in V(A_{{\mathcal{C}}^{*}})-V(B_{{\mathcal{C}}^{*}})$. So $v\in V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}$. Since there exists no $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex, $v$ is not a $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex. Since $y\in N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}})-(V(B_{{\mathcal{C}}^{*}})\cup S^{*})$, $y\not\in Y_{1}-T^{*}$. Since $y\in Y_{1}$, $y\in T^{*}$. So $y\in Y_{B}$, a contradiction. ∎

Define the following:

• •

For every $y\in Y_{B}$, let $L_{B}(y)$ be a 1-element subset of $L(y)$.

• •

For every $v\in V(G_{B})-Y_{B}$ with $\lvert N_{G_{B}}(v)\cap Y_{B}\rvert\in[s^{\prime}-1]$, let $L_{B}(v)$ be a subset of $L(v)$ with size $s^{\prime}+2-\lvert N_{G_{B}}(v)\cap Y_{B}\rvert$ such that $L_{B}(v)\cap L_{B}(u)=\emptyset$ for every $u\in N_{G}(v)\cap Y_{B}$. (Note that such a subset of $L(v)$ exists by 16.12.)

• •

For every other vertex $v$ of $G_{B}$, let $L_{B}(v):=L(v)$.

Hence $L_{B}$ is a restricted $(s^{\prime},2,Y_{B})$-list-assignment by 16.12. Since $({\mathcal{C}}^{*},S^{*},T^{*})$ is useful and $\iota^{*}(y)=y$ for every $y\in Y_{1}\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}*})$,

[TABLE]

by (U3).

Since $\lvert Y_{B}\rvert<\lvert Y_{1}\rvert$, we know $\lvert V(G_{B})\rvert<\lvert V(G)\rvert$. By the minimality of $\lvert V(G)\rvert$, there exists an $(\eta,g)$-bounded $L_{B}$-coloring $c_{B}$ of $G_{B}$. Define the following:

• •

Let $G_{A}:=G[V(A_{{\mathcal{C}}^{*}})\cup Z]$.

• •

Let $Y_{A}:=(Y_{1}\cap V(A_{{\mathcal{C}}^{*}}))\cup Z\cup S^{*}\cup T^{*}\cup V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})$.

• •

For every $y\in Y_{A}$, let $L_{A}(y)$ be a 1-element subset of $L(y)$ such that if $y\in V(G_{B})$, then $L_{A}(y)=\{c_{B}(y)\}$.

• •

For every $v\in V(G_{A})-Y_{A}$ with $1\leqslant\lvert N_{G_{A}}(v)\cap Y_{A}\rvert\leqslant s^{\prime}-1$, let $L_{A}(v)$ be a subset of $L(v)$ with size $s^{\prime}+2-\lvert N_{G_{A}}(v)\cap Y_{A}\rvert$ such that $L_{A}(v)\cap L_{A}(u)=\emptyset$ for every $u\in Y_{A}\cap N_{G_{A}}(v)$.

• •

For every other vertex $v$ of $G_{A}$, let $L_{A}(v):=L(v)$.

Then $L_{A}$ is a restricted $(s^{\prime},2,Y_{A})$-list-assignment of $G_{A}$. Since $\lvert{\mathcal{C}}^{*}\rvert\leqslant a_{(d-1)a_{0}}$, $\lvert V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\rvert\leqslant(d-1)a_{(d-1)a_{0}}<\theta-\xi$. So $(A_{{\mathcal{C}}^{*}},B_{{\mathcal{C}}^{*}})\in{\mathcal{T}}-Z$ and hence $\lvert Y_{1}\cap V(A_{{\mathcal{C}}^{*}})\rvert\leqslant 3\theta$. By (U3), $\lvert S^{*}\rvert+\lvert T^{*}\rvert+\lvert Z\rvert\leqslant\lvert Y_{1}\cap V(A_{{\mathcal{C}}^{*}})-V(B_{{\mathcal{C}}^{*}})\rvert+\lvert Y_{1}\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\rvert\leqslant\lvert Y_{1}\cap V(A_{{\mathcal{C}}^{*}})\rvert\leqslant 3\theta$. Hence $\lvert Y_{A}\rvert\leqslant\lvert Y_{1}\cap V(A_{{\mathcal{C}}^{*}})\rvert+\lvert S^{*}\rvert+\lvert T^{*}\rvert+\lvert Z\rvert+\lvert V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\rvert\leqslant 3\theta+3\theta+\theta\leqslant 7\theta$.

In particular, $\lvert V(G_{A})\rvert<\lvert V(G)\rvert$. By minimality, there exists an $(\eta,g)$-bounded $L_{A}$-coloring $c_{A}$ of $G_{A}$.

Claim 16.13**.**

For every $v\in V(G_{A})-(V(G_{B})\cup Y_{1})$ with $N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}\neq\emptyset$, $c_{B}(u)\not\in L_{A}(v)$ for every $u\in N_{G}(v)\cap V(G_{B})$.

Proof.

Since $v\in V(G_{A})-(V(G_{B})\cup Y_{1})$, $v\in V(G)-(Z\cup V(B_{{\mathcal{C}}^{*}}))$. So there exists $(A,B)\in{\mathcal{C}}^{*}$ such that $v\in V(A)-V(B)$. Hence $N_{G}(v)\subseteq V(A)$ and $\lvert N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\rvert\leqslant\lvert N_{G}(v)\cap V(A\cap B)\rvert\leqslant d-1$.

Since $v\in N_{G}(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})$ and ${\mathcal{C}}^{*}$ satisfies (U2), $v\not\in Z^{\prime}$. So $\lvert N_{G}(v)\cap Z\rvert\leqslant s-1$.

Since $N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}\neq\emptyset$, there exists $w\in N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}$. Since there exists no $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex, $w$ is not a $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex. Since $S^{*}\subseteq V(G_{B})$, $v\not\in S^{*}$. Since $v\not\in Y_{1}$, $v\not\in Y_{1}-T^{*}$. So $\lvert N_{G}(v)\cap((Y_{1}\cap V(A_{{\mathcal{C}}^{*}}))\cup(S^{*}-V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})))\rvert\leqslant 2d-5$.

Since $T^{*}\subseteq Y_{1}\cap V(A_{{\mathcal{C}}^{*}})$,

[TABLE]

So by the definition of $L_{A}$, $L_{A}(v)\cap\{c_{B}(u)\}=L_{A}(v)\cap L_{A}(u)=\emptyset$ for every $u\in Y_{A}\cap V(G_{B})\cap N_{G}(v)$. Since $N_{G}(v)\cap V(G_{B})\subseteq Y_{A}$, $c_{B}(u)\not\in L_{A}(v)$ for every $u\in N_{G}(v)\cap V(G_{B})$. ∎

Let $c$ be the $L$-coloring of $G$ defined by $c(v):=c_{A}(v)$ if $v\in V(G_{A})$, and $c(v):=c_{B}(v)$ if $v\in V(G)-V(G_{A})$.

Claim 16.14**.**

Let $M$ be a monochromatic $c$-component intersecting both $V(G_{A})-V(G_{B})$ and $V(G_{B})-V(G_{A})$. Then every component of $M\cap G_{A}$ intersects $Y_{A}$, and every component of $M\cap G_{B}$ intersects $Y_{B}$.

Proof.

Since $M$ intersects both $V(G_{A})-V(G_{B})$ and $V(G_{B})-V(G_{A})$, every component of $M\cap G_{A}$ intersects $V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})\cup Z\cup S^{*}\cup T^{*}\subseteq Y_{A}$.

Let $M_{B}$ be a component of $M\cap G_{B}$. Suppose that $M_{B}$ is disjoint from $Y_{B}=(Y_{1}\cap V(B_{{\mathcal{C}}^{*}}))\cup Z\cup S^{*}\cup T^{*}$. Since $M$ intersects both $V(G_{A})-V(G_{B})$ and $V(G_{B})-V(G_{A})$, there exist $u\in V(M_{B})\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}$ and $v\in N_{M_{B}}(v)\cap V(A_{{\mathcal{C}}^{*}})-(V(B_{{\mathcal{C}}^{*}})\cup S^{*}\cup T^{*})$. Since $u$ is not a $({\mathcal{C}}^{*},S^{*},T^{*})$-dangerous vertex, $v\not\in Y_{1}-T^{*}$. Since $v\not\in T^{*}$, $v\not\in Y_{1}$. So $v\in V(G_{A})-(V(G_{B})\cup Y_{1})$ and $N_{G}(v)\cap V(A_{{\mathcal{C}}^{*}}\cap B_{{\mathcal{C}}^{*}})-S^{*}\supseteq\{u\}\neq\emptyset$. By 16.13, $c(v)=c_{A}(v)\neq c_{B}(u)=c(u)$. But $M$ is a monochromatic $c$-component, a contradiction. Hence every component of $M\cap G_{B}$ intersects $Y_{B}$. ∎

Let $U_{A}$ be the union of the monochromatic $c_{A}$-components of $G_{A}$ intersecting $Y_{A}$. Let $U_{B}$ be the union of the monochromatic $c_{B}$-components of $G_{B}$ intersecting $Y_{B}$. Since $c_{A}$ and $c_{B}$ are $(\eta,g)$-bounded, $\lvert V(U_{A})\cup V(U_{B})\rvert\leqslant\lvert Y_{A}\rvert^{2}g(\lvert Y_{A}\rvert)+\lvert Y_{B}\rvert^{2}g(\lvert Y_{B}\rvert)\leqslant(7\theta)^{2}g(7\theta)+(\lvert Y_{1}\rvert-1)^{2}g(\lvert Y_{1}\rvert-1)\leqslant g(\lvert Y_{1}\rvert)$.

Since $V(G)\subseteq V(G_{A})\cup V(G_{B})$, by 16.14, every monochromatic $c$-component intersecting both $V(G_{A})-V(G_{B})$ and $V(G_{B})-V(G_{A})$ is contained in $U_{A}\cup U_{B}$ and hence contains at most $g(\lvert Y_{1}\rvert)\leqslant\eta^{2}g(\eta)$ vertices. Let $M$ be a monochromatic $c$-component. If $V(M)\subseteq V(G_{A})$, then $M$ is a monochromatic $c_{A}$-component with at most $\eta^{2}g(\eta)$ vertices since $c_{A}$ is $(\eta,g)$-bounded. If $V(M)\subseteq V(G_{B})$, then $M$ is a monochromatic $c_{B}$-component with at most $\eta^{2}g(\eta)$ vertices since $c_{B}$ is $(\eta,g)$-bounded. Hence every monochromatic $c$-component contains at most $\eta^{2}g(\eta)$ vertices.

Since $Y_{1}\subseteq Y_{A}\cup Y_{B}$, by 16.14, the union of the monochromatic $c$-components intersecting $Y_{1}$ is contained in $U_{A}\cup U_{B}$, so it contains at most $g(\lvert Y_{1}\rvert)\leqslant\lvert Y_{1}\rvert^{2}g(\lvert Y_{1}\rvert)$ vertices. Therefore, $c$ is an $(\eta,g)$-bounded $L$-coloring of $G$, a contradiction. This proves the theorem. ∎

4 Excluding Almost $(\leqslant 1)$-Subdivisions

Recall that an almost $(\leqslant 1)$-subdivision of a graph $H$ is a graph obtained from $H$ by subdividing edges such that at most one edge is subdivided more than once. The following simple observation is useful.

Lemma 17**.**

For $s\in\mathbb{N}$, let $G$ be a graph and let $H$ be a subgraph of $G$ isomorphic to $K_{s-1,t}$ for some $t\geqslant\binom{s-1}{2}+2$. Let $(X,Z)$ be the bipartition of $H$ with $\lvert X\rvert=s-1$. If $G$ does not contain an almost $(\leqslant 1)$-subdivision of $K_{s+1}$, then each component of $G-X$ contains at most one vertex in $Z$, and $G-X$ has at least two components.

Proof.

Let $C_{1},C_{2},\dots,C_{k}$ be the components of $G-X$. For each $i\in[k]$, $\lvert V(C_{i})\cap Z\rvert\leqslant 1$, as otherwise $G[X\cup Z]$ together with a path in $C_{i}$ connecting two vertices in $V(C_{i})\cap Z$ is an almost $(\leqslant 1)$-subdivision of $K_{s+1}$, a contradiction. Hence $k\geqslant|Z|\geqslant t\geqslant 2$. ∎

The following lemma shows that a result for graphs excluding a $K_{s,t}$ subgraph can be extended for graphs excluding an almost $(\leqslant 1)$-subdivision of $K_{s+1}$. Let $s,r\in\mathbb{N}$. Let $G$ be a graph and $Y_{1}\subseteq V(G)$. An $(s,r,Y_{1})$-list-assignment of $G$ is said to be faithful if for every $v\in V(G)-Y_{1}$ with $\lvert N_{G}(v)\cap Y_{1}\rvert=s$, we have $L(v)-\bigcup_{y\in Y_{1}\cap N_{G}(v)}L(y)\neq\emptyset$.

Lemma 18**.**

Let ${\mathcal{G}}$ be a subgraph-closed family of graphs. Let $\beta,r$ be functions with domain $\mathbb{N}$ such that $\beta(x)\geqslant x$ and $r(x)\in\mathbb{N}$ for every $x\in\mathbb{N}$.

Assume that for every $s\in\mathbb{N}$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every $G\in{\mathcal{G}}$ with no $K_{s,t_{s}}$ subgraph, where $t_{s}:=\max\{\binom{s}{2}+2,s+2\}$, for every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and for every $(\beta(s),r(s),Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$.

Then for every $s\in\mathbb{N}$ with $s\geqslant 2$, there exist $\eta^{*}\in\mathbb{N}$ and a nondecreasing function $g^{*}$ such that for every graph $G\in{\mathcal{G}}$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, for every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta^{*}$, and for every faithful $(\beta(s-1),r(s-1),Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta^{*},g^{*})$-bounded $L$-coloring of $G$.

Proof.

For every $s\in\mathbb{N}$, let $\eta_{s}$ be the number and $g_{s}$ be the function such that for every $G\in{\mathcal{G}}$ with no $K_{s,t_{s}}$ subgraph, every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta_{s}$ and every $(\beta(s),r(s),Y_{1})$-list-assignment of $G$, there exists an $(\eta_{s},g_{s})$-bounded $L$-coloring of $G$. For every $s\in\mathbb{N}$ with $s\geqslant 2$, let $\eta_{s}^{*}:=\eta_{s-1}$ and let $g_{s}^{*}$ be the function defined by $g_{s}^{*}(0):=g_{s-1}(0)$ and $g_{s}^{*}(x):=g_{s-1}(x)+\eta_{s}^{*}\cdot g_{s}^{*}(x-1)$ for every $x\in{\mathbb{N}}$.

Fix $s\in\mathbb{N}-\{1\}$. Let $\beta^{\prime}:=\beta(s-1)$ and $r^{\prime}:=r(s-1)$. We shall prove that for every graph $G$ in ${\mathcal{G}}$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, for every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta_{s}^{*}$, and for every faithful $(\beta^{\prime},r^{\prime},Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta_{s}^{*},g^{*}_{s})$-bounded $L$-coloring of $G$.

Suppose to the contrary that $G$ is a graph in ${\mathcal{G}}$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, $Y_{1}$ is a subset of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta^{*}_{s}$, and $L$ is a faithful $(\beta^{\prime},r^{\prime},Y_{1})$-list-assignment of $G$ such that there exists no $(\eta_{s}^{*},g_{s}^{*})$-bounded $L$-coloring of $G$. We further assume that $\lvert V(G)\rvert$ is as small as possible.

Since $\eta_{s}^{*}=\eta_{s-1}$ and $g_{s}^{*}\geqslant g_{s-1}$, there exists no $(\eta_{s-1},g_{s-1})$-bounded $L$-coloring of $G$. Since $\eta_{s}^{*}=\eta_{s-1}$, by the definition of $\eta_{s-1}$ and $g_{s-1}$, $G$ contains a $K_{s-1,t_{s-1}}$ subgraph. Let $t^{\prime}$ be the maximum integer such that $G$ contains a $K_{s-1,t^{\prime}}$ subgraph. So $t^{\prime}\geqslant t_{s-1}$. Let $H$ be a subgraph of $G$ isomorphic to $K_{s-1,t^{\prime}}$. Let $\{P,Q\}$ be the bipartition of $H$ such that $\lvert P\rvert=s-1$ and $\lvert Q\rvert=t^{\prime}$. By the maximality of $t^{\prime}$, $Q$ is the set of all vertices of $V(G)-P$ adjacent in $G$ to all vertices in $P$.

Claim 18.1**.**

Every component of $G-P$ contains some vertex in $Y_{1}$.

Proof.

Suppose to the contrary that there exists a component $C$ of $G-P$ disjoint from $Y_{1}$. By Lemma 17, $G-P$ contains at least two components and there exists at most one vertex in $C$ adjacent in $G$ to all vertices in $P$. By the minimality of $G$, there exists an $(\eta_{s}^{*},g_{s}^{*})$-bounded $L|_{V(G)-V(C)}$-coloring $c$ of $G-V(C)$.

Since $\beta^{\prime}\geqslant s-1$ and $L$ is an $(\beta^{\prime},r^{\prime},Y_{1})$-list-assignment and $V(C)\cap Y_{1}=\emptyset$, for every $v\in V(C)$ with $\lvert N_{G}(v)\cap Y_{1}\rvert\leqslant\beta^{\prime}-1$, we have $L(v)\cap\bigcup_{y\in N_{G}(v)\cap Y_{1}}L(y)=\emptyset$ and

[TABLE]

For every $v\in V(C)$ with $\lvert N_{G}(v)\cap Y_{1}\rvert\geqslant\beta^{\prime}$,

[TABLE]

implying $P\subseteq Y_{1}$ and $\lvert N_{G}(v)\cap Y_{1}\rvert=\beta^{\prime}$, so $L(v)-\{c(y):y\in N_{G}(v)\cap P\}=L(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}}L(y)\neq\emptyset$ since $L$ is faithful. So for every $v\in V(C)$, $L(v)-\{c(y):y\in N_{G}(v)\cap P\}\neq\emptyset$.

Let $L^{\prime}$ be the following list-assignment of $G[V(C)\cup P]$:

• •

For every $v\in P$, let $L^{\prime}(v):=\{c(v)\}$.

• •

For every $v\in V(C)$ with $\lvert N_{G}(v)\cap P\rvert\geqslant\beta^{\prime}$, let $L^{\prime}(v)$ be a 1-element subset of $L(v)-\bigcup_{u\in P}L^{\prime}(u)$.

• •

Let $Y_{1}^{\prime}:=P\cup\{v\in V(C):\lvert N_{G}(v)\cap P\rvert\geqslant\beta^{\prime}\}$.

• •

For every $v\in V(C)\cap N^{<\beta^{\prime}}(Y_{1}^{\prime})$, let $L^{\prime}(v)$ be a subset of $L(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}^{\prime}}L^{\prime}(y)$ of size $\lvert L(v)\rvert-\lvert N_{G}(v)\cap Y_{1}^{\prime}-Y_{1}\rvert=\beta^{\prime}+r^{\prime}-\lvert N_{G}(v)\cap Y_{1}^{\prime}\rvert\geqslant r^{\prime}+1$.

• •

For every $v\in V(C)\cap N^{<\beta^{\prime}}(P)-N^{<\beta^{\prime}}(Y_{1}^{\prime})$, let $L^{\prime}(v)$ be a subset of $L(v)-\bigcup_{y\in N_{G}(v)\cap P}L^{\prime}(y)$ of size $\lvert L(v)\rvert-\lvert N_{G}(v)\cap P-Y_{1}\rvert=\beta^{\prime}+r^{\prime}-\lvert N_{G}(v)\cap P\rvert\geqslant r^{\prime}+1$.

• •

For every $v\in V(C)-(Y_{1}^{\prime}\cup N^{<\beta^{\prime}}(Y_{1}^{\prime})\cup N^{<\beta^{\prime}}(P))$, let $L^{\prime}(v):=L(v)$.

Note that $Y_{1}^{\prime}-P$ consists of the vertex in $V(C)$ adjacent in $G$ to all vertices in $P$. Hence for every $v\in V(C)\cap N_{G}(P)-Y_{1}^{\prime}$, $\lvert N_{G}(v)\cap P\rvert\in[\beta^{\prime}-1]$. That is, $V(C)\cap N_{G}(P)-Y_{1}^{\prime}=V(C)\cap N^{<\beta^{\prime}}(P)$. So for every $v\in V(C)\cap N_{G}(P)-Y_{1}^{\prime}$, $L^{\prime}(v)\cap L^{\prime}(u)=\emptyset$ for every $u\in P\cap N_{G}(v)$.

Clearly, $L^{\prime}$ is an $(\beta^{\prime},r^{\prime},Y_{1}^{\prime})$-list-assignment of $G$. If $v\in V(C)-Y_{1}^{\prime}$ with $\lvert N_{G}(v)\cap Y_{1}^{\prime}\rvert=\beta^{\prime}$, then since $\lvert Y_{1}^{\prime}-P\rvert=1$, we know $v\in N^{<\beta^{\prime}}(P)-N^{<\beta^{\prime}}(Y_{1}^{\prime})$, so $L^{\prime}(v)$ is a set of size at least $r^{\prime}+1\geqslant 2$ disjoint from $\bigcup_{y\in N_{G}(v)\cap P}L^{\prime}(y)$. Hence if $v\in(V(C)\cup P)-Y_{1}^{\prime}$ with $\lvert N_{G}(v)\cap Y_{1}^{\prime}\rvert=\beta^{\prime}$, then $L^{\prime}(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}^{\prime}}L^{\prime}(y)=L^{\prime}(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}^{\prime}-P}L^{\prime}(y)$ has size $\lvert L^{\prime}(y)\rvert-1\geqslant 1$. Therefore, $L^{\prime}$ is a faithful $(\beta^{\prime},r^{\prime},Y_{1}^{\prime})$-list-assignment of $G[V(C)\cup P]$.

Since $G-P$ contains at least two components, $\lvert V(C)\cup P\rvert<\lvert V(G)\rvert$. By the minimality of $G$, there exists an $(\eta_{s}^{*},g_{s}^{*})$-bounded $L^{\prime}$-coloring $c^{\prime}$ of $G[V(C)\cup P]$.

For every $v\in V(C)\cap N_{G}(P)$, if $v\in N^{<\beta^{\prime}}(P)$, then $L^{\prime}(v)$ is disjoint from $\bigcup_{y\in N_{G}(v)\cap P}L^{\prime}(y)$; if $v\in V(C)$ with $\lvert N_{G}(v)\cap P\rvert\geqslant\beta^{\prime}$, then $v\in Y_{1}^{\prime}-P$ and $L^{\prime}(v)$ is disjoint from $\bigcup_{y\in P}L^{\prime}(y)$. Hence every monochromatic $c^{\prime}$-component intersecting $P$ is contained in $G[P]$.

Let $c^{*}$ be the $L$-coloring defined by $c^{*}(v):=c(v)$ if $v\in V(G)-V(C)$, and $c^{*}(v):=c^{\prime}(v)$ if $v\in V(C)$. Hence every monochromatic $c^{*}$-component is either contained in $G-V(C)$ or contained in $C$, so it contains at most ${\eta_{s}^{*}}^{2}g(\eta_{s}^{*})$ vertices. Since $V(C)\cap Y_{1}=\emptyset$, the union of the monochromatic $c^{*}$-components intersecting $Y_{1}$ equals the union of the monochromatic $c$-components intersecting $Y_{1}$, and contains at most $\lvert Y_{1}\rvert^{2}g(\lvert Y_{1}\rvert^{2})$ vertices. Therefore, $c^{*}$ is an $(\eta_{s}^{*},g_{s}^{*})$-bounded $L$-coloring of $G$, a contradiction. ∎

Let $C_{1},C_{2},\dots,C_{k}$ be the components of $G-P$. For $i\in[k]$, let $G_{i}:=G[V(C_{i})\cup P]$. By Lemma 17, $k\geqslant t_{s-1}\geqslant s+1$. By 18.1, $Y_{1}\cap V(C_{i})\neq\emptyset$ for each $i\in[k]$, so $\lvert V(G_{i})\cap Y_{1}\rvert\leqslant\lvert Y_{1}\rvert-(k-1)\leqslant\lvert Y_{1}\rvert-s$ for each $i\in[k]$, and $k\leqslant\lvert Y_{1}\rvert\leqslant\eta_{s}^{*}$. So $\lvert(V(G_{i})\cap Y_{1})\cup P\rvert\leqslant\lvert Y_{1}\rvert-s+\lvert P\rvert<\lvert Y_{1}\rvert$ for each $i\in[k]$.

Let $L^{*}$ be the following list-assignment of $G$:

• •

Let $Y_{1}^{*}:=Y_{1}\cup P$.

• •

For each $v\in Y_{1}^{*}$, let $L^{*}(v)$ be a 1-element subset of $L(v)$.

• •

For each $v\in N^{<\beta^{\prime}}(Y_{1}^{*})$, let $L^{*}(v)$ be a subset of $L(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}^{*}}L^{*}(y)$ with size $\lvert L(v)\rvert-\lvert N_{G}(v)\cap(Y_{1}^{*}-Y_{1})\rvert=\beta^{\prime}+r^{\prime}-\lvert N_{G}(v)\cap Y_{1}^{*}\rvert$.

• •

For each $v\in V(G)-(Y_{1}^{*}\cup N^{<\beta^{\prime}}(Y_{1}^{*}))$, let $L^{*}(v):=L(v)$.

Clearly, $L^{*}$ is an $(\beta^{\prime},r^{\prime},Y_{1}^{*})$-list-assignment of $G$. Let $v\in V(G)-Y_{1}^{*}$ with $\lvert N_{G}(v)\cap Y_{1}^{*}\rvert=\beta^{\prime}$. So $L^{*}(v)=L(v)$. If $\lvert N_{G}(v)\cap Y_{1}\rvert=\beta^{\prime}$, then $N_{G}(v)\cap Y_{1}^{*}=N_{G}(v)\cap Y_{1}$, so $L^{*}(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}^{*}}L^{*}(y)=L(v)-\bigcup_{y\in N_{G}(v)\cap Y_{1}}L(y)\neq\emptyset$ since $L$ is a faithful $(\beta^{\prime},r^{\prime},Y_{1})$-list-assignment of $G$. If $\lvert N_{G}(v)\cap Y_{1}\rvert<\beta^{\prime}$, then $\lvert L^{*}(v)\rvert=\lvert L(v)\rvert=\beta^{\prime}+r^{\prime}-\lvert N_{G}(v)\cap Y_{1}\rvert$ and $L(v)$ is disjoint from $\bigcup_{y\in N_{G}(v)\cap Y_{1}}L(y)$, so

[TABLE]

Therefore, $L^{*}$ is a faithful $(\beta^{\prime},r^{\prime},Y_{1}^{*})$-list-assignment of $G$.

Since $P\subseteq Y_{1}^{*}$, $L^{*}|_{V(G_{i})}$ is a faithful $(\beta^{\prime},r^{\prime},Y_{1}^{*}\cap V(G_{i}))$-list-assignment of $G_{i}$. Recall that for each $i\in[k]$, $\lvert Y_{1}^{*}\cap V(G_{i})\rvert\leqslant\lvert Y_{1}\rvert-1\leqslant\eta$. By the minimality of $G$, for each $i\in[k]$, there exists an $(\eta_{s}^{*},g_{s}^{*})$-bounded $L^{*}|_{V(G_{i})}$-coloring $c_{i}$ of $G_{i}$. Since $P\subseteq Y_{1}^{*}\cap V(G_{i})$ for every $i\in[k]$, we know for every $v\in P$, $c_{i}(v)=c_{j}(v)$ for any $i,j\in[k]$. Let $c^{*}$ be the $L^{*}$-coloring of $G$ defined by $c(v):=c_{1}(v)$ if $v\in P$, and $c(v):=c_{i}(v)$ if $v\in V(C_{i})$ for some $i\in[k]$.

Since $P\subseteq Y_{1}^{*}\cap V(G_{i})$ for all $i\in[k]$, the number of vertices in the union of the monochromatic $c^{*}$-components intersecting $Y_{1}\cup P$ is at most

[TABLE]

Furthermore, every monochromatic $c^{*}$-component disjoint from $Y_{1}\cup P$ is a monochromatic $c_{i}$-component for some $i\in[k]$, and hence contains at most ${\eta_{s}^{*}}^{2}g_{s}^{*}(\eta)$ vertices. Therefore $c^{*}$ is an $(\eta_{s}^{*},g_{s}^{*})$-bounded $L$-coloring of $G$. This proves the lemma. ∎

The following lemma is equivalent to Lemma 18 except it applies for restricted list assignments. The proof is identical, so we omit it.

Lemma 19**.**

Let ${\mathcal{G}}$ be a subgraph-closed family of graphs. Let $\beta,r$ be functions with domain $\mathbb{N}$ such that $\beta(x)\geqslant x$ and $r(x)\in\mathbb{N}$ for every $x\in\mathbb{N}$.

Assume that for every $s\in\mathbb{N}$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every $G\in{\mathcal{G}}$ with no $K_{s,t_{s}}$ subgraph, where $t_{s}:=\max\{\binom{s}{2}+2,s+2\}$, for every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$, and for every restricted $(\beta(s),r(s),Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$.

Then for every $s\in\mathbb{N}$ with $s\geqslant 2$, there exist $\eta^{*}\in\mathbb{N}$ and a nondecreasing function $g^{*}$ such that for every graph $G\in{\mathcal{G}}$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, for every $Y_{1}\subseteq V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta^{*}$, and for every restricted faithful $(\beta(s-1),r(s-1),Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta^{*},g^{*})$-bounded $L$-coloring of $G$.

Theorem 20**.**

If $s\in\mathbb{N}$ with $s\geqslant 2$, then the following hold:

For every $w\in\mathbb{N}$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every graph $G$ of treewidth at most $w$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, every subset $Y_{1}$ of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$ and every faithful $(s-1,1,Y_{1})$-list-assignment of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$. 2. 2.

For every graph $H$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every graph $G$ with no $H$-minor and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, every subset $Y_{1}$ of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$ and every restricted faithful $(s-1,2,Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$. 3. 3.

For every $d\in\mathbb{N}$ with $d\geqslant 2$ and graph $H$ of maximum degree at most $d$, there exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every graph $G$ with no $H$-subdivision and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, every subset $Y_{1}$ of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$ and every restricted faithful $(s+3d-7,2,Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$. 4. 4.

There exist $\eta\in\mathbb{N}$ and a nondecreasing function $g$ such that for every graph $G$ with no $K_{s+1}$-subdivision, every subset $Y_{1}$ of $V(G)$ with $\lvert Y_{1}\rvert\leqslant\eta$ and every restricted faithful $(4s-7,2,Y_{1})$-list-assignment $L$ of $G$, there exists an $(\eta,g)$-bounded $L$-coloring of $G$.

Proof.

Statement 1 follows from Theorems 8 and 18 by taking ${\mathcal{G}}$ to be the set of graphs of treewidth at most $w$, $\beta(s)=s$ and $r(s)=1$.

Statement 2 follows from Theorems 9 and 19 by taking ${\mathcal{G}}$ to be the set of graphs with no $H$-minor, $\beta(s)=s$ and $r(s)=2$.

Statement 3 follows from Lemmas 19 and 16 by taking ${\mathcal{G}}$ to be the set of graphs with no $H$-subdivision, $\beta(s)=3d+s-6$ and $r(s)=2$. Note that $\beta(s)\geqslant s$ since $d\geqslant 2$. And $3d+s\geqslant 7$ since $d\geqslant 2$ and $s\geqslant 1$.

Statement 4 follows from Statement 3 by taking $H=K_{s+1}$. ∎

When $Y_{1}=\emptyset$, every $(s,r,Y_{1})$-list-assignment is faithful. Thus, Theorem 20 implies that for all $s,d,w\in\mathbb{N}$ with $s\geqslant 2$ and $d\geqslant 2$, for every graph $H$, there exists $\eta\in\mathbb{N}$ such that:

For every graph $G$ with treewidth at most $w$ and with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, and for every list-assignment $L$ of $G$ with $\lvert L(v)\rvert\geqslant s$ for every $v\in V(G)$, there exists an $L$-coloring with clustering $\eta$ (Theorem 1 for $s\geqslant 2$). 2. 2.

For every graph $G$ with no almost $(\leqslant 1)$-subdivision of $K_{s+1}$ and with no $H$-minor, there exists an $(s+1)$-coloring of $G$ with clustering $\eta$ (Theorem 2 for $s\geqslant 2$). 3. 3.

If the maximum degree of $H$ is at most $d$, then for every graph $G$ with no $H$-subdivision and no almost $(\leqslant 1)$-subdivision of $K_{s+1}$, there exists an $(s+3d-5)$-coloring of $G$ with clustering $\eta$ (Theorem 4 for $s\geqslant 2$ and $d\geqslant 2$). 4. 4.

For every graph $G$ with no $K_{s+1}$-subdivision, there exists a $(4s-5)$-coloring of $G$ with clustering $\eta$ (Theorem 6 for $s\geqslant 2$).

Note that when $s=1$, graphs with no $K_{s+1}$ subgraph have no edge, so they are 1-colorable with clustering 1. This together with Lemma 15 complete the proof of Theorems 1, 2, 4 and 6.