Improper Colourings inspired by Hadwiger's Conjecture

TL;DR

This paper explores improper colourings of $K_t$-minor-free graphs, providing new bounds on colourability and monochromatic component sizes, and extends results to $K_{s,t}$-minor-free graphs and graphs with no $K_t$-immersion.

Contribution

It introduces novel improper colouring bounds for $K_t$-minor-free graphs with smaller monochromatic components and degrees, and extends these results to related graph classes.

Findings

$K_t$-minor-free graphs are $(2t-2)$-colourable with small monochromatic components.

$K_t$-minor-free graphs are $(t-1)$-colourable with bounded monochromatic degree.

Graphs with no $K_t$-immersion are 2-colourable with bounded monochromatic degree.

Abstract

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with monochromatic components of order at most $\lceil{\frac12(t-2)}\rceil$. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$-minor-free graph is $(t-1)$-colourable with monochromatic degree at most $t-2$. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for $K_{s,t}$-minor-free graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no $K_t$-immersion are $2$-colourable with bounded monochromatic degree.

Equations

$$|L|⩽|L^{\prime }|=|S_1|+|S_2|+|Z|⩽|S|+\frac{1}{2}|L\setminus S_1|⩽t+\frac{1}{2}|L|.$$

$${\mathrm{scol}}_r(G)⩽p(k+1)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽p\left(\genfrac{}{}{0ex}{}{r+k}{k}\right)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽\left(\genfrac{}{}{0ex}{}{t-1}{2}\right)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽(t-3)\left(\genfrac{}{}{0ex}{}{r+t-2}{t-2}\right)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽2(t-1)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽(t-1)(r+1)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽3(2t+1)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽(2t+1)\left(\genfrac{}{}{0ex}{}{r+2}{2}\right)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽3(4g+7)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽(4g+7)\left(\genfrac{}{}{0ex}{}{r+2}{2}\right)(2r+1).$$

$${\mathrm{wcol}}_r(G)⩽f(s,t)r^s.$$

$${\mathrm{scol}}_r(G)⩽s(s+1)(t-1)(2r+1)\text{and}{\mathrm{wcol}}_r(G)⩽s(t-1)\left(\genfrac{}{}{0ex}{}{r+s}{s}\right)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽3(2r+1).$$

$${\mathrm{scol}}_r(G)⩽(2g+3)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽(4g+6)(k+1)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽(2g+3)(2d+1)(2r+1).$$

$${\mathrm{scol}}_r(G)⩽4k(2r+1).$$

Full text

21st April 2017, revised 23rd February 2018

MSC: 05C15, 05C83

1 Introduction

Hadwiger’s Conjecture [32] asserts that every $K_{t}$-minor-free graph has a proper $(t-1)$-colouring. For $t\leqslant 3$ the conjecture is easy. Hadwiger [32] and Dirac [15] independently proved the conjecture for $t=4$; while Wagner’s result [58] means that the case $t=5$ is equivalent to the Four Colour Theorem. Finally, Robertson et al. [52] proved Hadwiger’s Conjecture for $t=6$. The conjecture remains open for $t\geqslant 7$. Hadwiger’s Conjecture is widely considered to be one of the most important open problems in graph theory. The best upper bound on the chromatic number of $K_{t}$-minor-free graphs is $\mathcal{O}(t\sqrt{\log t})$ independently due to Kostochka [41, 42] and Thomason [55, 56]. See the recent survey by Seymour [53] for more on Hadwiger’s Conjecture.

One possible way to approach Hadwiger’s Conjecture is to allow improper colourings. In a vertex-coloured graph, a monochromatic component is a connected component of the subgraph induced by all the vertices of one colour. A graph $G$ is $k$-colourable with clustering $c$ if each vertex can be assigned one of $k$ colours such that each monochromatic subgraph has at most $c$ vertices111This type of colouring is sometimes called “fragmented” in the literature, but we feel that increased fragmentation suggests smaller monochromatic components, hence we use the term “clustered”.. Kleinberg et al. [40] introduced this type of colouring, and now many results are known. The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ for which there exists an integer $c$ such that every graph in $\mathcal{G}$ is $k$-colourable with clustering $c$.

Kawarabayashi and Mohar [37] were the first to prove an $\mathcal{O}(t)$ upper bound on the clustered chromatic number of $K_{t}$-minor-free graphs. In particular, they proved that every $K_{t}$-minor-free graph is \bigl{\lceil}{\frac{31}{2}t}\bigr{\rceil}-colourable with clustering $f(t)$, for some function $f$. The number of colours in this result was improved to \bigl{\lceil}{\tfrac{1}{2}(7t-3)}\bigr{\rceil} by Wood [60] 222This result depends on a result announced in 2008 which is not yet written., to $4t-4$ by Edwards et al. [23], and to $3t-3$ by Liu and Oum [45]. See [36, 35] for analogous results for graphs excluding odd minors. For all of these results, the function $f(t)$ is very large, often depending on constants from the Graph Minor Structure Theorem [50].

Our first contribution is to prove an analogous theorem with the best known number of colours, and also with small clustering. The proof is simple, and does not depend on any deep theory.

Theorem 1**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph is $(2t-2)$-colourable with clustering \bigl{\lceil}{\tfrac{1}{2}(t-2)}\bigr{\rceil}.

Theorem 1 implies that the clustered chromatic number of $K_{t}$-minor-free graphs is at most $2t-2$. A construction of Edwards et al. [23] mentioned below implies that the clustered chromatic number of $K_{t}$-minor-free graphs is at least $t-1$.

A second way to relax the conclusion in Hadwiger’s Conjecture is to bound the maximum degree of monochromatic components. A graph $G$ is $k$-colourable with defect $d$ if each vertex can be assigned one of $k$ colours such that each vertex is adjacent to at most $d$ vertices of the same colour; that is, each monochromatic subgraph has maximum degree at most $d$. Cowen et al. [9] introduced the notion of defective graph colouring, and now many results for various graph classes are known. A graph class $\mathcal{G}$ is defectively $k$-colourable if there exists an integer $d$ such that every graph in $\mathcal{G}$ is $k$-colourable with defect $d$. The defective chromatic number of $\mathcal{G}$ is the minimum integer $k$ such that $\mathcal{G}$ is defectively $k$-colourable [10]. Edwards et al. [23] proved that every $K_{t}$-minor-free graph is $(t-1)$-colourable with defect $\mathcal{O}(t^{2}\log t)$. Moreover, it is shown in [23] that the number of colours, $t-1$, is best possible in the following strong sense: for every integer $d$, there is a $K_{t}$-minor-free graph that is not $(t-2)$-colourable with defect $d$. Thus the defective chromatic number of $K_{t}$-minor-free graphs equals $t-1$. (This also shows that the clustered chromatic number of $K_{t}$-minor-free graphs is at least $t-1$.)

Our second contribution is an improved upper bound on the defect in the result of Edwards et al. [23].

Theorem 2**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph is $(t-1)$-colourable with defect $t-2$.

Edwards et al. [23] wisely noted that their theorem mentioned earlier should not be considered evidence for the truth of Hadwiger’s Conjecture, since their method also proves that every $K_{t}$-topological-minor-free graph is $(t-1)$-colourable with defect $\mathcal{O}(t^{4})$. It is not true that every $K_{t}$-topological-minor-free graph is properly $(t-1)$-colourable. This last statement is Hajós’ Conjecture, which is now known to be false [5, 57]. On the other hand, our proof of Theorem 2 does not work for graphs excluding a topological minor.

Theorems 1 and 2 are corollaries of the following decomposition result of independent interest. A sequence $H_{1},\dots,H_{\ell}$ is a connected partition of a graph $G$ if each $H_{i}$ is a non-empty connected induced subgraph of $G$, the subgraphs $H_{1},\dots,H_{\ell}$ are pairwise disjoint, and $V(G)=V(H_{1})\cup\dots\cup V(H_{\ell})$. Two disjoint subgraphs $H$ and $H^{\prime}$ of a graph $G$ are adjacent if there is an edge in $G$ with one endpoint in $H$ and one endpoint in $H^{\prime}$. For a positive integers $n,m$, we use $[n]$ to denote the set $\{1,\ldots,n\}$ and $[n,m]$ to denote the set $\{n,\ldots,m\}$.

Theorem 3**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ such that for $i\in[\ell]$:

1. (1)

$H_{i}$* is adjacent to at most $t-2$ of the subgraphs $H_{1},\dots,H_{i-1}$;* 2. (2)

$H_{i}$* has maximum degree at most $t-2$; and* 3. (3)

$H_{i}$* is $2$-colourable with clustering \bigl{\lceil}{\tfrac{1}{2}(t-2)}\bigr{\rceil}.*

We actually prove a decomposition theorem with several further properties; see Theorem 11. It is easy to derive Theorems 2 and 1 from Theorem 3. Colour the subgraphs $H_{1},\dots,H_{\ell}$ greedily in this order, such that adjacent subgraphs receive distinct colours. By property (1), $t-1$ colours suffice. Theorem 2 follows from property (2) by colouring each vertex in $H_{i}$ by the colour assigned to $H_{i}$. Theorem 1 follows from property (3) by taking the product of the $(t-1)$-colouring of $H_{1},\dots,H_{\ell}$ with the given $2$-colouring of each subgraph $H_{i}$.

Theorem 3 is an extension of a result by Van den Heuvel et al. [34] in which properties (2) and (3) are replaced by “$H_{i}$ has a Breadth-First Search (BFS) spanning tree with at most $t-3$ leaves”. Van den Heuvel et al. [34] were motivated by connections to generalised colouring numbers. Note that the result in [34] implies that $H_{i}$ has at most $t-3$ vertices in each BFS layer. It follows that the maximum degree of $H_{i}$ is at most $3t-10$. Alternately colouring the BFS layers shows that $H_{i}$ is $2$-colourable with clustering $t-3$. Constructing $H_{i}$ more carefully, and choosing the $2$-colouring more carefully, leads to the improved bounds in Theorem 3, which we prove in Section 3.

Our main decomposition theorem, Theorem 11, also has the following corollary, which might be of independent interest.

Theorem 4**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ such that:

1. (1)

the quotient graph $Q$ obtained by contracting each $H_{i}$ to a single vertex is chordal with clique size at most $t-1$ (and hence has treewidth at most $t-2$); and 2. (2)

each part $H_{i}$ has bandwidth (and hence pathwidth and treewidth) at most $t-3$.

Hadwiger’s Conjecture implies that for every graph $H$ with $t$ vertices, the maximum chromatic number of $H$-minor-free graphs equals $t-1$ (since $K_{t-1}$ is $H$-minor-free). However, for clustered and defective colourings, fewer colours often suffice. For example, it follows from the main result by Ossona de Mendez et al. [47] that for every fixed non-complete graph $H$ on $t$ vertices, every $H$-minor-free graph is $(t-2)$-colourable with bounded defect, which is one fewer colour than in the complete graph case. More interestingly,, Archdeacon [3] proved that graphs embeddable in a fixed surface are defectively $3$-colourable (see also [61, 8, 10, 9]); while Dvořák and Norin [21] proved that such graphs are $4$-colourable with bounded clustering. Ossona de Mendez et al. [47] conjectured that for every connected graph $H$, the defective chromatic number of $H$-minor-free graphs equals the treedepth of $H$ minus $1$. They proved this conjecture for $K_{s,t}$-minor-free graphs, by showing that they are defectively $s$-colourable. Note that $K_{3,t}$-minor-free graphs are of particular interest since they include and generalise graphs embeddable in fixed surfaces. In the case $s\leqslant 3$ we prove decomposition results analogous to Theorem 3 that imply this result of [47] with much improved bounds on the defect. This direction is explored in Section 4.

In the same way as Van den Heuvel et al. [34] applied their decomposition result for $K_{t}$-minor-free graphs to the setting of generalised colouring numbers, we apply our decomposition results for $K_{s,t}$-minor-free graphs and $K^{*}_{s,t}$-minor-free graphs (where $K^{*}_{s,t}$ is the complete join of $K_{s}$ and $\overline{K_{t}}$) to conclude new bounds on generalised colouring numbers. Our results when specialised for graphs of given genus are almost as strong as the best known bounds. We then show how the concept of layered treewidth also leads to good bounds on generalised colouring numbers. The advantage of this approach is that it immediately applies to several non-minor-closed graph classes of interest. These results on generalised colouring numbers are presented in Section 5.

The final section, Section 6, returns to the topic of defective graph colouring, but instead of excluding a $K_{t}$ minor we exclude a $K_{t}$ immersion. The analogue of Hadwiger’s Conjecture, that $K_{t}$-immersion-free graphs are properly $(t-1)$-colourable [44, 1], is open. For defective colouring, we show that only 2 colours suffice.

Before continuing, we mention an important connection between clustered and defective colourings, implicitly observed in [23]. We include the proof for completeness.

Lemma 5** (Edwards et al. [23]).**

For every minor-closed graph class $\mathcal{G}$, the clustered chromatic number of $\,\mathcal{G}$ is at most three times the defective chromatic number of $\,\mathcal{G}$.

Proof.

Liu and Oum [45] proved that for every minor-closed graph class $\mathcal{G}$ and integer $d$, there is an integer $c=c(\mathcal{G},d)$ such that every graph in $\mathcal{G}$ with maximum degree $d$ is $3$-colourable with clustering $c$. (Esperet and Joret [24] previously proved an analogous result for graphs on surfaces.) Let $k$ be the defective chromatic number of $\mathcal{G}$. Thus for some integer $d$, every graph $G$ in $\mathcal{G}$ is $k$-colourable with defect $d$. Apply the result of Liu and Oum [45] to each monochromatic component of $G$, which has maximum degree at most $d$. Then $G$ is $3k$-colourable with clustering $c$, and hence the clustered chromatic number of $\mathcal{G}$ is at most $3k$. ∎

2 Preliminaries

2.1 Notation and definitions

This subsection briefly states standard graph theoretic definitions probably familiar to most readers.

A graph $H$ is a minor of a graph $G$ if a graph isomorphic to $H$ can be obtained from a subgraph of $G$ by contracting edges. Equivalently, and often easier to use intuitively: a graph $H$ with vertices $v_{1},\ldots,v_{n}$ is a minor of $G$ if there exist pairwise disjoint connected subgraphs $H_{1},\ldots,H_{n}$ of $G$ such that for every edge $v_{i}v_{j}$ in $H$, $H_{i}$ and $H_{j}$ are adjacent in $G$. We call $H_{i}$ the branch set corresponding to $v_{i}$. A class of graphs $\mathcal{G}$ is minor-closed if for every graph $G\in\mathcal{G}$, every minor of $G$ is also in $\mathcal{G}$. A graph $H$ is a topological minor of a graph $G$ if a graph isomorphic to a subdivision of $H$ is a subgraph of $G$.

The Euler genus of an orientable surface with $h$ handles is $2h$. The Euler genus of a non-orientable surface with $c$ cross-caps is $c$. The Euler genus of a graph $G$ is the minimum Euler genus of a surface in which $G$ embeds (with no crossing edges).

A tree decomposition of a graph $G$ is given by a tree $T$ whose nodes index a collection$(\,T_{x}\subseteq V(G)\mid x\in V(T)\,)$ of sets of vertices in $G$ called bags, such that (1) for every edge $vw$ of $G$, some bag $T_{x}$ contains both $v$ and $w$, and (2) for every vertex $v$ of $G$, the set $\{\,x\in V(T)\mid v\in T_{x}\,\}$ induces a non-empty (connected) subtree of $T$. The width of a tree decomposition $T$ is \max\{\,|T_{x}\bigm{|}x\in V(T)\,\}-1, and the treewidth of a graph $G$ is the minimum width of the tree decompositions of $G$. Note that the treewidth of $G$ equals the minimum integer $k$ such that $G$ is a subgraph of a chordal graph with clique number $k+1$.

A path decomposition is a tree decomposition in which the underlying tree is a path. The pathwidth of a graph $G$ is the minimum width of a path decomposition of $G$.

For a graph $G$ and $A,B\subseteq V(G)$, an $AB$-separator is a set $S\subseteq V(G)$ such that every $AB$-path in $G$ contains a vertex from $S$. (Note that we allow $A$ and $B$ to intersect and that all vertices in $A\cap B$ must be included in any $AB$-separator.) A pair $(G_{1},G_{2})$ is a $k$-separation of a graph $G$ if $G_{1}$ and $G_{2}$ are induced subgraphs of $G$ such that $G=G_{1}\cup G_{2}$, $G_{1}\not\subseteq G_{2}$ and $G_{2}\not\subseteq G_{1}$, and $|V(G_{1})\cap V(G_{2})|=k$.

2.2 Connected Induced Subgraphs

This subsection contains some elementary results about connected induced subgraphs containing a given set of vertices. We look in detail at so-called Lexicographic Breadth-First Search (LexBFS) trees, since these form a key tool in our methods.

A layering of a graph $G$ is a partition $(V_{0},V_{1},\dots,V_{\ell})$ of $V(G)$ such that for every edge $vw\in E(G)$, if $v\in V_{i}$ and $w\in V_{j}$, then $|i-j|\leqslant 1$. Each set $V_{i}$ is called a layer.

Let $r$ be a vertex in a connected graph $G$. Let $\ell=\max\{\,\operatorname{dist}_{G}(r,v)\mid v\in V(G)\,\}$, and for $i\in\{0,\ldots,\ell\}$ define $V_{i}=\{\,v\in V(G)\mid\operatorname{dist}_{G}(r,v)=i\,\}$. Then $V_{0},V_{1},\dots,V_{\ell}$ is a layering of $G$, called the BFS layering of $G$ starting from the root $r$; each $V_{i}$ is called a BFS layer of $G$. A spanning tree $T$ of $G$ rooted at $r$ is a BFS spanning tree if $\operatorname{dist}_{G}(v,r)=\operatorname{dist}_{T}(v,r)$ for every vertex $v$ in $G$. A BFS subtree is a subtree of a BFS spanning tree that includes the root. Let $S$ be a BFS subtree rooted at $r$ and consider a vertex $v\in V_{i}\cap V(S)$ for some $i\geqslant 1$. Let $P_{v}$ be the $vr$-path in $S$. Then $P_{v}$ has exactly one vertex in each of $V_{0},\dots,V_{i}$. The parent of $v$ is the neighbour of $v$ (in $S$) in $V_{i-1}$. Every vertex $x$ in $G$ is adjacent to at most three vertices in $P_{v}$ (since if $x\in V_{j}$, then $N_{G}(x)\subseteq V_{j-1}\cup V_{j}\cup V_{j+1}$). A leaf in a rooted tree is a non-root vertex of degree 1. If $S$ has $p$ leaves, then every vertex in $G$ is adjacent to at most $3p$ vertices in $S$. This observation can be improved for a special type of BFS (sub)trees.

For our purposes, a BFS spanning tree $T$ of $G$ is a LexBFS spanning tree if each BFS layer $V_{i}$ can be linearly ordered such that:

1. (a)

each vertex $v\in V_{i}$ with parent $w\in V_{i-1}$ in $T$ has no neighbour in $G$ that comes before $w$ in the ordering of $V_{i-1}$ (called the priority rule); and 2. (b)

for every edge $vw$ in $T$ with $v\in V_{i}$ and $w\in V_{i-1}$, there is no edge $xy$ in $T$ with $x$ before $v$ in the ordering of $V_{i}$ and $y$ after $w$ in the ordering of $V_{i-1}$ (called the non-crossing rule).

It is easily seen that every connected graph has a LexBFS spanning tree rooted at any given vertex. A LexBFS subtree is a subtree of a LexBFS spanning tree that includes the root.

Throughout this paper we follow the convention that the root of a rooted tree (such as a BFS or LexBFS (sub)tree) is never a leaf.

Lemma 6**.**

For $k\geqslant 1$, if $S$ is a LexBFS subtree of a connected graph $G$ and $S$ has $k$ leaves, then every vertex in $G$ has at most $2k$ neighbours in $V(S)$.

Proof.

Let $T$ be a LexBFS spanning tree of $G$, such that $S$ is a subtree of $T$. Let $V_{0},\dots,V_{\ell}$ be the BFS layers of $T$. Let $v$ be a vertex in $V_{i}$ (which may or may not be in $S$). If $v$ is on some leaf-root path $P$ of $S$, then $|N_{G}(v)\cap V(P)|\leqslant 2$. Now consider a leaf-root path $P$ in $S$ not containing $v$. Suppose on the contrary that there are distinct vertices $x,y,z\in N_{G}(v)\cap V(P)$, none of which are on a leaf-root path of $T$ containing $v$. Without loss of generality, $x\in V_{i-1}$, $y\in V_{i}$ and $z\in V_{i+1}$. Let $w$ be the parent of $v$ in $T$. So $w\in V_{i-1}$, but $w\neq x$ (since $x$ is not on a leaf-root path of $T$ containing $v$). By the priority rule, $w$ comes before $x$ in the ordering of $V_{i-1}$. By the non-crossing rule, $v$ comes before $y$ in the ordering of $V_{i}$, which contradicts the priority rule for $z$. Thus $|N_{G}(v)\cap V(P)|\leqslant 2$. Since there are $k$ leaf-root paths in $S$, in total this gives $|N_{G}(v)\cap V(S)|\leqslant 2k$. ∎

A graph $G$ has bandwidth at most $k$ if there is a vertex ordering $v_{1},\dots,v_{n}$ of $V(G)$, such that $|i-j|\leqslant k$ for each edge $v_{i}v_{j}$ of $G$.

Lemma 7**.**

Every connected graph $G$ that has a LexBFS spanning tree $T$ with $k$ leaves has bandwidth, pathwidth and treewidth at most $k$.

Proof.

Say $T$ is rooted at $r$. Let $V_{0},\dots,V_{\ell}$ be the BFS layers of $T$. Each $V_{i}$ is linearly ordered by LexBFS. We claim that the vertex-ordering of $V(G)$ produced by using the orderings of $V_{0},\dots,V_{\ell}$ in that order has bandwidth at most $k$. Consider an edge $vw$ where $v\in V_{i}$ and $w\in V_{i}$. Since $T$ has at most $k$ leaves, $|V_{i}|\leqslant k$ and at most $k-2$ vertices are between $v$ and $w$ in $V_{0},\dots,V_{\ell}$. Now consider an edge $vw$ where $v\in V_{i}$ and $w\in V_{i+1}$. Let $X$ be the set of vertices that come after $v$ in $V_{i}$ or come before $w$ in $V_{i+1}$. Then $X$ is the set of vertices between $v$ and $w$ in the ordering of $V(G)$. Let $p$ be the parent of $w$ in $T$. By the priority rule, $p\notin X$. By the non-crossing rule, no vertex in $X\cup\{v\}$ is a descendent of another vertex in $X\cup\{v\}$. Hence, the number of leaves in $T$ is at least $|X|+1$, implying $|X|\leqslant k-1$. Therefore $G$ has bandwidth at most $k$.

It is well-known and easy to prove that the pathwidth of a graph is at most its bandwidth (and hence so is the treewidth). Take the vertex ordering $v_{1},\dots,v_{n}$ of $V(H)$ that shows $H$ has bandwidth at most $k$. For $i\in[n-k]$, let $T_{i}=\{v_{i},\ldots,v_{i+k}\}$. Then $T_{1},T_{2},\dots,T_{n-k}$ defines the desired path decomposition. ∎

Lemma 8**.**

For every set $A$ of $\,k\geqslant 2$ vertices in a connected graph $G$, every minimal induced connected subgraph $H$ of $\,G$ containing $A$ satisfies the following properties:

1. (1)

every (non-rooted) subtree of $\,H$ has at most $k$ leaves; 2. (2)

$H$* has maximum degree at most $k$;* 3. (3)

$H$* has bandwidth (and hence pathwidth and treewidth) at most $k-1$;* 4. (4)

$H$* can be $2$-coloured with clustering \bigl{\lceil}{\tfrac{1}{2}k}\bigr{\rceil}; and* 5. (5)

$H$* can be $2$-coloured with $\{\text{red},\text{blue}\}$ such that there are at most $k-2$ red vertices and the blue subgraph consists of at most $k-1$ pairwise disjoint paths.*

Proof.

Let $T$ be a spanning tree of $H$. By the minimality of $H$, every leaf of $T$ is in $A$. Thus $T$ has at most $k$ leaves. Now let $S$ be any tree in $H$. Extending $S$ to a spanning tree of $H$ cannot decrease the number of leaves, hence $S$ also has at most $k$ leaves.

The closed neighbourhood of a vertex $v\in V(H)$ contains a tree with $\deg_{H}(v)$ leaves, proving $\deg_{H}(v)\leqslant k$.

Let $T$ be a LexBFS spanning tree of $H$ rooted at a vertex $r$ in $A$. By the minimality of $H$, every leaf of $T$ is in $A$. Thus $T$ has at most $k-1$ leaves (the root does not count as a leaf). By Lemma 7, $H$ has bandwidth, pathwidth and treewidth at most $k-1$.

We now prove (4). We proceed by induction on $|V(H)|$. In the base case, $|V(H)|=|A|=k$ and the result is trivial. Now assume that $|V(H)|>k$. Thus $V(H-A)\neq\varnothing$, and by the minimality of $H$, every vertex in $H-A$ is a cut-vertex of $H$. Consider a leaf-block $L$ of $H$. Every vertex in $L$, except the one cut-vertex in $L$, is in $A$. There are at least two leaf-blocks. Thus $|V(L-v)|\leqslant\tfrac{1}{2}k$ for some leaf block $L$, where $v$ is the one cut-vertex of $H$ in $L$. Let $H^{\prime}=H-V(L-v)$ and $A^{\prime}=(A\setminus V(L))\cup\{v\}$. Then $H^{\prime}$ is a minimal induced connected subgraph of $G$ containing $A^{\prime}$, and $|A^{\prime}|\leqslant k$. By induction, $H^{\prime}$ has a $2$-colouring with clustering \bigl{\lceil}{\tfrac{1}{2}k}\bigr{\rceil}. Colour every vertex in $L\setminus\{v\}$ by the colour not assigned to $v$ in $H^{\prime}$. Now $H$ is $2$-coloured with clustering \bigl{\lceil}{\tfrac{1}{2}k}\bigr{\rceil}.

It remains to prove (5). We proceed by induction on $k$. If $k=2$, then $H$ is a path between the two vertices in $A$. Colour every vertex in $H$ blue, and we are done. So assume $k\geqslant 3$ and the result holds for $k-1$. Let $x$ be a vertex in $A$. By induction, every minimal induced connected subgraph $H^{\prime}$ of $H$ containing $A\setminus\{x\}$ can be $2$-coloured with $\{\text{red},\text{blue}\}$ such that there are at most $k-3$ red vertices and the blue subgraph consists of at most $k-2$ pairwise disjoint paths. If $x$ is in $H^{\prime}$, then we are done. Otherwise, let $P$ be a shortest path between $x$ and $H^{\prime}$ in $H$. Say $P=x,\dots,u,v,w$, where $w$ is in $H^{\prime}$. Then $v$ is the only vertex in $P-w$ adjacent to $H^{\prime}$. Colour $v$ red, and colour $x,\dots,u$ blue. (It is possible that $x=v$, in which case $\{x,\dots,u\}=\varnothing$.) Then $\{x,\dots,u\}$ induces a path in $H$ that is not adjacent to $H^{\prime}$. By the minimality of $H$, we have $V(H)=V(H^{\prime})\cup\{x,\dots,u,v\}$. Thus $H$ is $2$-coloured with $\{\text{red},\text{blue}\}$ such that there are at most $k-2$ red vertices and the blue subgraph consists of at most $k-1$ pairwise disjoint paths. ∎

We now prove the main result of this section.

Lemma 9**.**

For every set $A$ of $\,k\geqslant 2$ vertices in a connected graph $G$, there is an induced connected subgraph $H$ of $\,G$ containing $A$, such that:

1. (1)

$H$* has maximum degree at most $k$;* 2. (2)

$H$* has bandwidth (and hence pathwidth and treewidth) at most $k-1$;* 3. (3)

$H$* can be $2$-coloured with clustering \bigl{\lceil}{\tfrac{1}{2}k}\bigr{\rceil};* 4. (4)

$H$* can be $2$-coloured with $\{\text{red},\text{blue}\}$ such that there are at most $k-2$ red vertices and the blue subgraph consists of at most $k-1$ pairwise disjoint paths; and* 5. (5)

every vertex in $G$ has at most $2k-2$ neighbours in $V(H)$.

Proof.

Let $T$ be a LexBFS spanning tree of $G$ rooted at some vertex $r\in A$. Let $S$ be the LexBFS subtree of $T$ consisting of all $ar$-paths in $T$, where $a\in A$. Every leaf of $S$ is in $A\setminus\{r\}$, implying that $S$ has at most $k-1$ leaves. By Lemma 6, every vertex in $G$ has at most $2k-2$ neighbours in $V(S)$. Let $H$ be a minimal induced connected subgraph of $G[V(S)]$ containing $A$. The first four claims follow from Lemma 8. Since $V(H)\subseteq V(S)$, Lemma 6 means that every vertex in $G$ has at most $2k-2$ neighbours in $V(H)$. ∎

3 Decompositions of $K_{t}$-Minor-Free

Graphs

Van den Heuvel et al. [34] introduced the following definition and proved the following decomposition theorem. A connected partition $H_{1},\dots,H_{\ell}$ has width $k$ if for each $i\in[\ell-1]$, each component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} is adjacent to at most $k$ of the subgraphs $H_{1},\dots,H_{i}$. Note that this implies that $H_{i+1}$ is adjacent to at most $k$ of the subgraphs $H_{1},\dots,H_{i}$ (since $H_{i+1}$ is contained in some component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}).

Theorem 10** (Van den Heuvel et al. [34]).**

Every $K_{t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $t-2$, such that each subgraph $H_{i}$ is induced by a BFS subtree of \,G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $t-3$ leaves.

The following similar decomposition theorem implies Theorem 3.

Theorem 11**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $t-2$, such that for $i\in[\ell]$ the following holds.

1. (1)

The subgraph $H_{i}$ has the following properties:

1. (a)

$H_{i}$* has maximum degree at most $t-2$;* 2. (b)

$H_{i}$* has bandwidth, pathwidth and treewidth at most $t-3$;* 3. (c)

$H_{i}$* can be $2$-coloured with clustering \bigl{\lceil}{\tfrac{1}{2}(t-2)}\bigr{\rceil}; and* 4. (d)

$H_{i}$* can be $2$-coloured with $\{\text{red},\text{blue}\}$ such that there are at most $t-4$ red vertices and the blue subgraph consists of at most $t-3$ pairwise disjoint paths.* 2. (2)

Each component $C$ of \,G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} has the following properties.

1. (a)

At most $t-2$ subgraphs in $H_{1},\dots,H_{i}$ are adjacent to $C$, and these subgraphs are pairwise adjacent. (This implies that at most $t-2$ subgraphs in $H_{1},\dots,H_{i}$ are adjacent to $H_{i+1}$, and these subgraphs are pairwise adjacent.) 2. (b)

Every vertex in $C$ is adjacent to at most $2t-6$ vertices in each of $\,H_{1},\dots,H_{i}$. (This implies that every vertex in $H_{i+1}$ is adjacent to at most $2t-6$ vertices in each of $\,H_{1},\dots,H_{i}$.)

Proof.

We may assume that $G$ is connected. We construct $H_{1},\dots,H_{\ell}$ iteratively, maintaining properties (1) and (2). Let $H_{1}$ be the subgraph induced by a single vertex in $G$. Then (1) and (2) hold for $i=1$.

Assume that $H_{1},\dots,H_{i}$ satisfy (1) and (2) for some $i\geqslant 1$, but $V(H_{1}),\dots,V(H_{i})$ do not partition $V(G)$. Let $C$ be a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}. Let $Q_{1},\dots,Q_{k}$ be the subgraphs in $H_{1},\dots,H_{i}$ that are adjacent to $C$. By (2a), $Q_{1},\dots,Q_{k}$ are pairwise adjacent and $k\leqslant t-2$. Since $G$ is connected, $k\geqslant 1$.

For $j\in[k]$, let $v_{j}$ be a vertex in $C$ adjacent to $Q_{j}$. If $k=1$, then let $H_{i+1}$ be the subgraph induced by $v_{1}$. It is easily seen that (1) is satisfied. For $k\geqslant 2$, by Lemma 9 with $k\leqslant t-2$, there is an induced connected subgraph $H_{i+1}$ of $C$ containing $v_{1},\dots,v_{k}$ that satisfies (1).

Consider a component $C^{\prime}$ of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)}. Either $C^{\prime}$ is disjoint from $C$, or $C^{\prime}$ is contained in $C$. If $C^{\prime}$ is disjoint from $C$, then $C^{\prime}$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} and $C^{\prime}$ is not adjacent to $H_{i+1}$, implying (2) is maintained for $C^{\prime}$.

Now assume $C^{\prime}$ is contained in $C$. Since every vertex in $C$ has at most $2t-6$ neighbours in each of $H_{1},\dots,H_{i}$, every vertex in $C^{\prime}$ has at most $2t-6$ neighbours in each of $H_{1},\dots,H_{i}$. By Lemma 9 (5), every vertex in $C^{\prime}$ also has at most $2t-6$ neighbours in $H_{i+1}$. Thus (2b) is maintained for $C^{\prime}$. The subgraphs in $H_{1},\dots,H_{i+1}$ that are adjacent to $C^{\prime}$ are a subset of $Q_{1},\dots,Q_{k},H_{i+1}$, which are pairwise adjacent. Suppose that $k=t-2$ and $C^{\prime}$ is adjacent to all of $Q_{1},\dots,Q_{t-2},H_{i+1}$. Then $C$ is adjacent to all of $Q_{1},\dots,Q_{t-2}$. Contracting each of $Q_{1},\dots,Q_{t-2},H_{i+1},C^{\prime}$ into a single vertex gives $K_{t}$ as a minor of $G$, a contradiction. Hence $C^{\prime}$ is adjacent to at most $t-2$ of $Q_{1},\dots,Q_{t-2},H_{i+1}$, and property (2a) is maintained for $C^{\prime}$. ∎

Property (1d) in Theorem 11, along with a greedy $(t-1)$-colouring of the subgraphs $H_{1},\dots,H_{\ell}$, gives the following results.

Theorem 12**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph has a $(2t-2)$-colouring such that for $t-1$ colours each monochromatic component has at most $t-4$ vertices, and for the other $t-1$ colours each monochromatic component is a path.

Corollary 13**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph has a $(3t-3)$-colouring such that for $t-1$ colours, each monochromatic component has at most $t-4$ vertices, and the other $2t-2$ colour classes are independent sets.

The same greedy $(t-1)$-colouring of the subgraphs $H_{1},\dots,H_{\ell}$, together with Theorem 11 (1b), gives the following result.

Theorem 14**.**

For $t\geqslant 4$, every $K_{t}$-minor-free graph has a $(t-1)$-colouring such that each monochromatic component has treewidth at most $t-3$.

Note that DeVos et al. [12] proved that for every proper minor-closed class of graphs, every graph in that class has a $2$-colouring such that each monochromatic component has bounded treewidth. Their proof again uses the Graph Minor Structure Theorem, leading to a very large bound on the treewidth.

Property (2a) in Theorem 11 means that if $Q$ is the graph obtained $G$ by contracting each subgraph $H_{i}$ to a single vertex, then $Q$ is chordal with no $K_{t}$-subgraph, and thus with treewidth at most $t-2$. Indeed, $H_{1},\dots,H_{\ell}$ defines an elimination ordering of $Q$. In the language of Reed and Seymour [49], $H_{1},\dots,H_{\ell}$ is a chordal decomposition with touching pattern $Q$. We only need that $Q$ is $(t-2)$-degenerate for Theorems 1 and 2, but it is interesting that, in fact, $Q$ has treewidth at most $t-2$.

Even though we do not use it explicitly in this paper, it is an interesting aspect of our decomposition that the superstructure (that is, $Q$) has bounded treewidth, as does each piece of the decomposition. There are several other properties in Theorem 11 we do not use, but we mention them since they might be useful for other applications.

4 Excluding a Complete Bipartite Minor

This section presents decomposition results analogous to Theorem 11 for $K_{s,t}$-minor-free graphs, leading to bounds on the defective and clustered chromatic number. Those decomposition results and the more technical proofs can be found towards the end of the section.

In fact, for most of this section we will consider the larger classes of $K^{*}_{s,t}$-minor-free graphs, where $K^{*}_{s,t}$ is the complete join of $K_{s}$ and $\overline{K_{t}}$. We start with $s\in\{1,2,3\}$ before considering the general case. Graphs with no $K_{1,t}$ minor (note that $K_{1,t}=K^{*}_{1,t}$) are easily coloured. Every such graph has maximum degree at most $t-1$, and is therefore $1$-colourable with defect $t-1$. Moreover, every BFS layer has at most $t-1$ vertices, so alternately colouring the BFS layers gives a $2$-colouring with clustering $t-1$.

Next consider the $s=2$ case. Ossona de Mendez et al. [47] proved that every $K^{*}_{2,t}$-minor-free graph is $2$-colourable with defect $\mathcal{O}(t^{3})$. Our decomposition results imply the following improvement.

Theorem 15**.**

Every $K^{*}_{2,t}$-minor-free graph is $2$-colourable with defect $2t-2$.

The decomposition results for $K^{*}_{2,t}$-minor-free graphs also imply that every is $4$-colourable with clustering $t-1$. This result can be improved as follows. The proof is inspired by a method of Gonçalves [29].

Theorem 16**.**

Every $K^{*}_{2,t}$-minor-free graph $G$ is $3$-colourable with clustering $t-1$. Moreover, for each edge $vw$ of $\,G$, there is such a $3$-colouring in which $v$ and $w$ are both isolated in their respective monochromatic subgraphs.

Now consider $K^{*}_{3,t}$-minor-free graphs. Ossona de Mendez et al. [47] proved that the defective chromatic number of $K^{*}_{3,t}$-minor-free graphs equals 3. In particular, every $K^{*}_{3,t}$-minor-free graph is $3$-colourable with defect $\mathcal{O}(t^{4})$. Our decomposition results again imply an improvement.

Theorem 17**.**

Every $K^{*}_{3,t}$-minor-free graph is $3$-colourable with defect $4t$, and is $6$-colourable with clustering $2t$.

It follows from Euler’s Formula that graphs with Euler genus $g$ exclude $K_{3,2g+3}$ as a minor. Thus the second part of Theorem 17 is related to the results of Esperet and Ochem [25] and Kawarabayashi and Thomassen [38] that every graph of Euler genus $g$ can be $5$-coloured with clustering $\mathcal{O}(g)$. Kleinberg et al. [40] constructed planar graphs that cannot be $3$-coloured with bounded clustering. We conjecture that every $K_{3,t}$-minor-free graph is $4$-colourable with clustering $f(t)$, for some function $f$.

It is possible to improve the bound on the cluster size for the $6$-colouring result in Theorem 17. In a $K^{*}_{3,t}$-minor-free graph, every BFS layer induces a $K^{*}_{2,t}$-minor-free graph, which is $3$-colourable with clustering $t-1$ by Theorem 16. Using disjoint sets of three colours for alternate BFS layers gives a $6$-colouring with clustering $t-1$.

Finally, in this section we consider general $K_{s,t}$-minor-free graphs. Ossona de Mendez et al. [47] proved that the defective chromatic number of $K_{s,t}$-minor-free graphs equals $s$. We show that the clustered chromatic number of $K_{s,t}$-minor-free graphs is at least $s+1$, thus generalising the above-mentioned lower bound of Kleinberg et al. [40].

Proposition 18**.**

For every $s\geqslant 1$, $t\geqslant\max\{s,3\}$ and $c\geqslant 1$, there is a $K_{s,t}$-minor-free graph $G_{s}$ such that every $s$-colouring of $\,G_{s}$ has a monochromatic component of order greater than $c$.

Proof.

Define $G_{s}$ recursively as follows. Let $G_{1}$ be the path on $c+1$ vertices. For $s\geqslant 2$, let $G_{s}$ be the graph obtained from $c$ disjoint copies of $G_{s-1}$ by adding one dominant vertex.

We claim that $G_{s}$ is not $s$-colourable with clustering $c$. We prove this claim by induction on $s\geqslant 1$. Obviously, $G_{1}$ is not $1$-colourable with clustering $c$. Now assume that $s\geqslant 2$ and $G_{s-1}$ is not $(s-1)$-colourable with clustering $c$. Suppose that $G_{s}$ has an $s$-colouring with clustering $c$. Say the dominant vertex in $G_{s}$ is coloured black. At most $c-1$ copies of $G_{s-1}$ contain a black vertex, which implies that at least one copy has no black vertex. Thus $G_{s-1}$ has an $(s-1)$-colouring with clustering $c$, which is a contradiction. Hence $G_{s}$ is not $s$-colourable with clustering $c$, as claimed.

It remains to show that $G_{s}$ is $K_{s,t}$-minor-free with $t\geqslant\max\{s,3\}$. We do so by induction on $s\geqslant 1$. $G_{1}$ is a path, and therefore contains no $K_{1,3}$ minor. $G_{2}$ is outerplanar, and therefore contains no $K_{2,3}$ minor. $G_{3}$ is planar, and therefore contains no $K_{3,3}$ minor.

Now assume that $s\geqslant 4$ and $G_{s-1}$ contains no $K_{s-1,s-1}$ minor, but $G_{s}$ contains a $K_{s,s}$ minor. Let $v$ be the dominant vertex in $G_{s}$. We may assume that $v$ is the entire image of one vertex in the $K_{s,s}$ minor in $G$. Since $K_{s,s}$ is $2$-connected, the $K_{s,s}$ minor is contained in one copy of $G_{s-1}$ plus $v$. Deleting any one vertex from $K_{s,s}$ gives a subgraph that contains a $K_{s-1,s-1}$ subgraph. Thus $G_{s-1}$ contains a $K_{s-1,s-1}$ minor, which is a contradiction. We conclude that for $s\geqslant 4$, $G_{s}$ has no $K_{s,s}$ minor, so certainly no $K_{s,t}$ minor with $t\geqslant s$ ($=\max\{s,3\}$). ∎

Determining the clustered chromatic number of $K_{s,t}$-minor-free graphs is an open problem. Proposition 18 provides a lower bound of $s+1$. Since $K_{s,t}$-minor-free graphs are defectively $s$-colourable [47], Lemma 5 implies an upper bound of $3s$. In general, for every graph $H$, it is possible that the clustered chromatic number of $H$-minor-free graphs is at most one more than the defective chromatic number of $H$-minor-free graphs.

We now give the structural results and proofs of the above statements in this section. All the results in this section are based on LexBFS, so we first present the following general lemma. Recall the definition of the width of a connected partition from the beginning of Section 3.

Lemma 19**.**

Suppose that a graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $k$. If each subgraph $H_{i}$ is induced by a BFS subtree of \,G-V\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $p$ leaves, then $G$ is $(k+1)$-colourable with defect $3p-1$, and $G$ is $(2k+2)$-colourable with clustering $p$.

If, in addition, each subgraph $H_{i}$ is induced by a LexBFS subtree of \,G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $p$ leaves, then $G$ is $(k+1)$-colourable with defect $2p$.

Proof.

Colour the subgraphs $H_{1},\dots,H_{\ell}$ greedily in this order, such that adjacent subgraphs receive distinct colours. Since the partition has width $k$, $k+1$ colours suffice. Colour each vertex in $H_{i}$ by the colour assigned to $H_{i}$. In each subgraph $H_{i}$ each BFS layer has at most $p$ vertices. Since a vertex in a BFS subtree has neighbours in its own layer and in the two layers below and above its own layer only, $H_{i}$ has maximum degree at most $3p-1$. Hence $G$ is $(k+1)$-colourable with defect $3p-1$. Moreover, if each subgraph $H_{i}$ is induced by a LexBFS subtree of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $p$ leaves, then by Lemma 6, $H_{i}$ has maximum degree $2p$.

For the clustering claim, alternately $2$-colour the BFS layers in each $H_{i}$, and take the product with the $(k+1)$-colouring of $H_{1},\dots,H_{\ell}$ to produce a $(2k+2)$-colouring of $G$ with clustering $p$. ∎

As an aside, note that Van den Heuvel et al. [34] proved that every planar graph has a connected partition $H_{1},\dots,H_{n}$ with width 2, such that each subgraph $H_{i}$ is a shortest path in G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)}. Thus, Lemma 19 with $k=2$ and $p=1$ implies that planar graphs are $3$-colourable with defect 2, which is the best possible result for defective $3$-colouring of planar graphs, first proved by Cowen et al. [9]. In fact, each monochromatic component is a path, which was previously proved by Goddard [28] and Poh [48].

For $K^{*}_{2,t}$-minor-free graphs we have the following.

Lemma 20**.**

Every $K^{*}_{2,t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $1$, such that each subgraph $H_{i}$ is induced by a LexBFS subtree of \,G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $t-1$ leaves.

Proof.

We may assume that $G$ is connected. We construct $H_{1},\dots,H_{\ell}$ iteratively. Let $H_{1}$ be the subgraph induced by a single vertex in $G$.

Assume that $H_{1},\dots,H_{i}$ are defined for some $i\geqslant 1$, and $C$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} adjacent to one of $H_{1},\dots,H_{i}$. (Since $G$ is connected, $C$ is adjacent to at least one of those subgraphs.) So $C$ is adjacent to $H_{a}$, for some $a\in[i]$, and to no other subgraph in $H_{1},\dots,H_{i}$; let $A$ be the set of vertices in $C$ adjacent to $H_{a}$, and let $r$ be a vertex in $A$. Let $S$ be a LexBFS subtree of $C$ rooted at $r$, such that every vertex in $A$ is in $S$, and subject to this property, $|V(S)|$ is minimal. Thus every leaf of $S$ is in $A$. Let $S_{0}$ be the subtree of $S$ obtained by deleting the leaves. If $S$ has at least $t$ leaves, then a $K^{*}_{2,t}$ minor is obtained by contracting $H_{a}$ to a vertex and contracting $S_{0}$ to a vertex. Thus $S$ has at most $t-1$ leaves. Let $H_{i+1}$ be the subgraph of $C$ induced by $V(S)$. Since every vertex in $A$ is in $S$, every component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)} is adjacent to at most one of $H_{1},\dots,H_{i+1}$. Iterating this process gives the desired partition. ∎

Lemmas 19 and 20 immediately imply Theorem 15 and show that $K^{*}_{2,t}$-minor-free graphs are $4$-colourable with clustering $t-1$. As expressed in Theorem 16, this can be improved to a $3$-colouring with the same clustering bounds, as we now prove.

Proof of Theorem 16..

We proceed by induction on $|V(G)|$. The claim is trivial if $|V(G)|\leqslant t+1$. Now assume that $vw$ is an edge in a $K^{*}_{2,t}$-minor-free graph $G$, and the result holds for $K^{*}_{2,t}$-minor-free graphs with fewer vertices than $G$. If $\deg_{G}(v)=1$, then by induction $G-v$ has a $3$-colouring in which $w$ is isolated in its monochromatic subgraph. Assign $v$ a colour not assigned to $w$. We obtain the desired colouring of $G$.

Now assume that $\deg_{G}(v)\geqslant 2$ and, similarly, $\deg_{G}(w)\geqslant 2$. Let $A$ and $B$ be disjoint sets of vertices in $G$ such that $v\in A$ and $w\in B$, $G[A]$ and $G[B]$ are connected, and $vw$ is the only edge between $A$ and $B$, and subject to these properties, $|A\cup B|$ is maximum. The sets $A$ and $B$ are well-defined, since $A=\{v\}$ and $B=\{w\}$ satisfy the conditions. Let $Z$ be the set of vertices in $V(G)\setminus(A\cup B)$ adjacent to both $A$ and $B$, and let $Y=V(G)\setminus(A\cup B\cup Z)$.

If $|Z|\geqslant t$, then contracting $A$ and $B$ into single vertices gives a $K^{*}_{2,t}$ minor. Thus $|Z|\leqslant t-1$. Since $G[A]$ is connected and every vertex in $Z$ is adjacent to $A$, $G[A\cup Z]$ is connected. Similarly, $G[B\cup Z]$ is connected.

Let $G_{1}$ be obtained from $G$ by contracting $G[B\cup Z]$ into a single vertex $x$. Note that $vx$ is an edge of $G_{1}$. Let $G_{2}$ be obtained from $G$ by contracting $G[A\cup Z]$ into a single vertex $y$. Note that $wy$ is an edge of $G_{2}$. Since $G_{1}$ and $G_{2}$ are minors of $G$, they both contain no $K^{*}_{2,t}$ minor. Since $\deg_{G}(v)\geqslant 2$ and $\deg_{G}(w)\geqslant 2$, both $G_{1}$ and $G_{2}$ have fewer vertices than $G$.

By induction, $G_{1}$ is $3$-colourable with clustering $t-1$ such that $v$ and $x$ are both isolated in their respective monochromatic subgraphs, and $G_{2}$ is $3$-colourable with clustering $t-1$ such that $w$ and $y$ are both isolated in their respective monochromatic subgraphs. Permute the colours in $G_{2}$ so that $x\in V(G_{1})$ and $y\in V(G_{2})$ receive the same colour, and $v\in V(G_{1})$ and $w\in V(G_{2})$ receive distinct colours.

Let $G_{3}$ be obtained from $G$ by contracting $G[A\cup B\cup Z]$ into a single vertex $z$. Note that $V(G_{3})=Y\cup\{z\}$. By induction, $G_{3}$ is $3$-colourable with clustering $t-1$ such that $z$ is isolated in its colour class. Permute the colours in $G_{3}$ so that $z$ receives the same colour as $x\in V(G_{1})$, which is the same colour assigned to $y\in V(G_{2})$.

Colour each vertex in $Z$ by the colour assigned to $x$ and $y$. Colour each vertex in $A$ by its colour in $G_{1}$. Colour each vertex in $B$ by its colour in $G_{2}$. Finally, colour each vertex in $Y$ by its colour in $G_{3}$.

Since $x$ is isolated in its monochromatic subgraph in $G_{1}$, $y$ is isolated in its monochromatic subgraph in $G_{2}$, and $z$ is isolated in its monochromatic subgraph in $G_{3}$, every monochromatic component intersecting $Z$ is contained in $Z$, and thus has at most $t-1$ vertices. Since $vw$ is the only edge between $A$ and $B$, and $v$ and $w$ are assigned distinct colours, every monochromatic component that intersects $A$ is contained in $A$, and therefore by induction has at most $t-1$ vertices. Similarly, every monochromatic component that intersects $B$ is contained in $B$, and therefore by induction has at most $t-1$ vertices. ∎

The following lemma is used in our decomposition result for $K^{*}_{3,t}$-minor-free graphs.

Lemma 21**.**

For every connected graph $G$, non-empty sets $A,B\subseteq V(G)$, and integer $t\geqslant 1$,

1. (1)

$G$* has a LexBFS subtree $T$ with at most $2t+1$ leaves, such that $T$ intersects both $A$ and $B$, and $V(T)$ separates $A$ and $B$; or* 2. (2)

$G$* has a $K_{1,t}$ minor with every branch set intersecting both $A$ and $B$.*

Proof.

Let $r$ be a vertex in $A$. Let $X$ be a LexBFS spanning tree of $G$ rooted at $r$. For a set $L\subseteq V(G)$, let $T_{L}$ be the subtree of $X$ consisting of the union of all paths in $X$ between $L$ and $r$. Choose $L\subseteq V(G)$ so that $V(T_{L})$ is an $AB$-separator, and subject to this property, $|L|$ is minimum. This is well-defined since if $L=V(G)$, then $V(T_{L})=V(G)$. By the minimality of $|L|$, every vertex in $L$ is a leaf of $T_{L}$. And by the definition of $T_{L}$, every leaf of $T_{L}$ is in $L$.

For each $x\in L$, let $p_{x}$ be the vertex closest to $x$ in $T_{L}$, such that $\deg_{T}(p_{x})\geqslant 3$ or $p_{x}=r$. Let $Q_{x}$ be the path in $T_{L}$ between $x$ and $p_{x}$ not including $p_{x}$. We call $Q_{x}$ the leaf path at $x$. Let $T_{0}=T_{L}-\bigcup_{x\in L}V(Q_{x})$. Let $H$ be the graph obtained from $G-V(T_{0})$ by contracting the leaf path $Q_{x}$ corresponding to each $x\in L$ into a single vertex $y_{x}$. We consider $A$ to also be a set of vertices in $H$, where a vertex $y_{x}$ is in $A$ if any vertex of $Q_{x}$ is in $A$, and similarly for $B$. Let $S$ be a minimum $AB$-separator in $H$.

First suppose that $|S|\geqslant t+1$. By Menger’s Theorem, there are $t+1$ pairwise disjoint $AB$-paths $Z_{1},\dots,Z_{t+1}$ in $H$. Since $V(T_{L})$ is an $AB$-separator in $G$, each $Z_{i}$ contains $y_{x}$ for some $x\in L$, and each vertex $y_{x}$ is on at most one path $Z_{i}$. For $i\in[t]$, if $Z_{i}$ contains $y_{x}$, then contract $Z_{i}\cup Q_{x}$ into a single vertex. If $Z_{t+1}$ contains $y_{x}$, then contract $Z_{t+1}\cup Q_{x}\cup T_{0}$ into a single vertex. We obtain a $K_{1,t}$ minor with every branch set intersecting both $A$ and $B$ (since each $Q_{x}$ is adjacent to $T_{0}$), and (2) holds.

Now assume that $|S|\leqslant t$. Let $S_{1}$ be the set of vertices $x\in L$ such that $y_{x}$ is in $S$. Let $S_{2}$ be the set of vertices in $G-S_{1}$ that correspond to vertices in $S$. Thus $|S|=|S_{1}|+|S_{2}|$. Let $Z$ be the set of vertices $z\in V(T_{0})$ such that $z=p_{x}$ for some $x\in L\setminus S_{1}$, and $z\neq p_{x}$ for all $x\in S_{1}$. Let $T^{\prime}=T_{L^{\prime}}$, where $L^{\prime}=S_{1}\cup S_{2}\cup Z$. Since $S$ separates $A$ and $B$ in $H$, and $T^{\prime}$ contains $T_{0}\cup S_{1}\cup S_{2}$ along with $Q_{x}$ for each $x\in S_{1}$, it follows that $V(T^{\prime})$ separates $A$ and $B$ in $G$.

By the definition of $p_{x}$, for each vertex $z\in Z$ there are at least two vertices $x$ and $x^{\prime}$ in $L\setminus S_{1}$ for which $z=p_{x}=p_{x^{\prime}}$. Thus $|L\setminus S_{1}|\geqslant 2|Z|$. By the choice of $L$, we can argue

[TABLE]

Hence $|L|\leqslant 2t$. Thus $T_{L}$ is a LexBFS subtree with at most $2t$ leaves, such that $V(T_{L})$ separates $A$ and $B$. Let $T$ be obtained from $T_{L}$ by adding a shortest path in $X$ from $T_{L}$ to $B$. Then $T$ is a LexBFS subtree with at most $2t+1$ leaves, such that $T$ intersects both $A$ and $B$, and $V(T)$ separates $A$ and $B$. ∎

We are now ready to prove the following structural lemma.

Lemma 22**.**

Every $K^{*}_{3,t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $2$, such that each subgraph $H_{i}$ is induced by a LexBFS subtree of \,G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)} with at most $2t+1$ leaves.

Proof.

We again may assume that $G$ is connected. We construct $H_{1},\dots,H_{\ell}$ iteratively, maintaining the property that for each $i\in[\ell-1]$, each component $C$ of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} is adjacent to at most two of $H_{1},\dots,H_{i}$, and if $C$ is adjacent to $H_{a}$ and $H_{b}$, for some distinct $a,b\in[i]$, then $H_{a}$ and $H_{b}$ are adjacent. Call this property $(\star)$.

Assume that $H_{1},\dots,H_{i}$ is defined, and $C$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}. (Hence $C$ satisfies property $(\star)$.)

Suppose $C$ is adjacent to $H_{a}$, for some $a\in[i]$, and to no other subgraph in $H_{1},\dots,H_{i}$. Let $H_{i+1}$ be a subgraph of $C$ induced by one vertex adjacent to $H_{a}$. Let $C^{\prime}$ be a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)}. If $C^{\prime}$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}, then $C^{\prime}$ is not adjacent to $H_{i}$, and $(\star)$ is maintained for $C^{\prime}$. Otherwise $C^{\prime}$ is a component of $C-V(H_{i+1})$, and $C^{\prime}$ is adjacent to $H_{i+1}$ and possibly $H_{a}$. Since $H_{i+1}$ and $H_{a}$ are adjacent, $(\star)$ holds for $C^{\prime}$.

Now assume that $C$ is adjacent to $H_{a}$ and $H_{b}$, for some distinct $a,b\in[i]$, and to no other subgraph in $H_{1},\dots,H_{i}$. Let $A$ be the set of vertices in $C$ adjacent to $H_{a}$, and let $B$ be the set of vertices in $C$ adjacent to $H_{b}$. By Lemma 21 above we have: (1) $C$ has a LexBFS subtree $T$ separating $A$ and $B$, such that $T$ intersects both $A$ and $B$, and $T$ has at most $2t+1$ leaves, or (2) $C$ has a $K_{1,t}$ minor with every branch set intersecting both $A$ and $B$. In case (1), let $H_{i+1}$ be the subgraph of $C$ induced by $V(T)$. Since $T$ intersects both $A$ and $B$, the subgraph $H_{i+1}$ is adjacent to both $H_{a}$ and $H_{b}$. Let $C^{\prime}$ be a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)}. If $C^{\prime}$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}, then $C^{\prime}$ is not adjacent to $H_{i+1}$, and $(\star)$ is maintained for $C^{\prime}$. Otherwise, $C^{\prime}$ is a component of $C-V(H_{i+1})$. Then $C^{\prime}$ is adjacent to $H_{i+1}$ and at most one of $H_{a}$ and $H_{b}$ (since $V(T)$ separates $A$ and $B$). Thus property $(\star)$ holds for $C^{\prime}$ (since $H_{i+1}$ is adjacent to both $H_{a}$ and $H_{b}$).

In Case (2), with $H_{a}$ and $H_{b}$ we obtain a $K^{*}_{3,t}$ minor in $G$, which is a contradiction. ∎

5 Generalised Colouring Numbers

This section presents bounds on generalised colouring numbers, first introduced by Kierstead and Yang [39]. Generalised colouring numbers are important because they characterise bounded expansion classes [62], they characterise nowhere dense classes [30], and have several algorithmic applications such as the constant-factor approximation algorithm for domination number by Dvořák [19], and the almost linear-time model-checking algorithm of Grohe et al. [31]. They also interpolate between degeneracy and treewidth (strong colouring numbers) and between degeneracy and treedepth (weak colouring numbers). See [34, 43, 46] for more details.

For a graph $G$, linear ordering $\preccurlyeq$ of $V(G)$, vertex $v\in V(G)$, and integer $r\geqslant 1$, let $S_{r}(G,\preccurlyeq,v)$ be the set of vertices $x\in V(G)$ for which there is a path $v=w_{0},w_{1},\dots,w_{r^{\prime}}=x$ of length $r^{\prime}\in[0,r]$ such that $x\preccurlyeq v$ and $v\prec w_{i}$ for all $i\in[r-1]$. Similarly, let $W_{r}(G,\preccurlyeq,v)$ be the set of vertices $x\in V(G)$ for which there is a path $v=w_{0},w_{1},\dots,w_{r^{\prime}}=x$ of length $r^{\prime}\in[0,r]$ such that $x\preccurlyeq v$ and $x\prec w_{i}$ for all $i\in[r^{\prime}-1]$. For a graph $G$ and integer $r\geqslant 1$, the $r$-strong colouring number $\operatorname{scol}_{r}(G)$ of $G$ is the minimum integer $k$ such that there is a linear ordering $\preccurlyeq$ of $V(G)$ with $|S_{r}(G,\preccurlyeq,v)|\leqslant k$ for each vertex $v$ of $G$. Similarly, the $r$-weak colouring number $\operatorname{wcol}_{r}(G)$ is the minimum integer $k$ such that there is a linear ordering $\preccurlyeq$ of $V(G)$ with $|W_{r}(G,\preccurlyeq,v)|\leqslant k$ for each vertex $v$ of $G$.

The following lemma is implicitly proved by Van den Heuvel et al. [34].

Lemma 23** (Van den Heuvel et al. [34]).**

Let $H_{1},\dots,H_{\ell}$ be a connected partition of a graph $G$ with width $k$, such that there exists $p$ such that for $i\in[\ell]$, $V(H_{i})=V(P_{i,1})\cup\dots\cup V(P_{i,p_{i}})$, where $p_{i}\leqslant p$ and each $P_{i,j}$ is a shortest path in G-\Bigl{(}\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)}\cup\bigl{(}V(P_{i,1})\cup\dots\cup V(P_{i,j-1})\bigr{)}\Bigr{)}. Then the generalised colouring numbers of $\,G$ satisfy for every $r\geqslant 1$:

[TABLE]

Note that the conditions on the paths $P_{i,j}$ in the lemma are implied if $H_{i}$ is induced by a BFS subtree with at most $p$ leaves in G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)}.

For example, combining Lemma 23 with a variant of Theorem 10, Van den Heuvel et al. [34] proved that every $K_{t}$-minor-free graph $G$ satisfies:

[TABLE]

Lemmas 23 and 20 imply:

Theorem 24**.**

For every $K^{*}_{2,t}$-minor-free graph $G$ and every $r\geqslant 1$,

[TABLE]

And Lemmas 23 and 22 imply:

Theorem 25**.**

For every $K^{*}_{3,t}$-minor-free graph $G$ and every $r\geqslant 1$,

[TABLE]

Since graphs with Euler genus $g$ exclude $K_{3,2g+3}$ as a minor, Theorem 25 implies that for every graph $G$ with Euler genus $g$,

[TABLE]

These result are within a constant factor of the best known bounds for graphs of Euler genus $g$, due to Van den Heuvel et al. [34]. Note that Theorem 25 applies to a broader class of graphs than those with bounded Euler genus. For example, the disjoint union of $g+1$ copies of $K_{5}$ has Euler genus $g+1$, but contains no $K_{3,3}$ minor. It is easy to construct $3$-connected examples as well.

We conjecture that Theorems 24 and 25 can be generalised as follows:

Conjecture 26**.**

There exists a function $f$ such that for every $K^{*}_{s,t}$-minor-free graph $G$ and every $r\geqslant 1$,

[TABLE]

26 would be implied by Lemma 23 and the following conjecture.

Conjecture 27**.**

For all $t\geqslant s\geqslant 1$, there exists an integer $p$, such that every $K^{*}_{s,t}$-minor-free graph $G$ has a connected partition $H_{1},\dots,H_{\ell}$ with width $s-1$, such that for $i\in[\ell]$, $V(H_{i})=V(P_{i,1})\cup\dots\cup V(P_{i,p_{i}})$, where $p_{i}\leqslant p$ and each $P_{i,j}$ is a shortest path in G-\Bigl{(}\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)}\cup\bigl{(}V(P_{i,1})\cup\dots\cup V(P_{i,j-1})\bigr{)}\Bigr{)}.

We now show that 26 is true with $r^{s}$ replaced by $r^{s+1}$.

Proposition 28**.**

For every $K^{*}_{s,t}$-minor-free graph $G$ and every $r\geqslant 1$, we have

[TABLE]

Proposition 28 follows from Lemma 23 and the next lemma.

Lemma 29**.**

Every $K^{*}_{s,t}$-minor-free graph has a connected partition $H_{1},\dots,H_{\ell}$ with width $s$, such that for $i\in[\ell]$, $V(H_{i})=V(P_{i,1})\cup\dots\cup V(P_{i,p_{i}})$, where $p_{i}\leqslant s(t-1)$ and each $P_{i,j}$ is a shortest path in G-\Bigl{(}\bigl{(}V(H_{1})\cup\dots\cup V(H_{i-1})\bigr{)}\cup\bigl{(}V(P_{i,1})\cup\dots\cup V(P_{i,j-1})\bigr{)}\Bigr{)}.

Proof.

Once more we may assume that $G$ is connected. We construct $H_{1},\dots,H_{\ell}$, maintaining the property that for each $i\in[\ell-1]$, each component $C$ of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} is adjacent to at most $s$ subgraphs in $H_{1},\dots,H_{i}$, and that the subgraphs $C$ is adjacent to are also pairwise adjacent. Call this property $(\star)$.

Assume that $H_{1},\dots,H_{i}$ is defined, and $C$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}. (Hence $C$ satisfies property $(\star)$.) Let $Q_{1},\dots,Q_{k}$ be the subgraphs in $H_{1},\dots,H_{i}$ that are adjacent to $C$. Thus $Q_{1},\dots,Q_{k}$ are pairwise adjacent and $k\leqslant s$.

Since $G$ is connected, $k\geqslant 1$. For $j\in[k]$, let $A_{j}$ be the set of vertices in $C$ adjacent to $Q_{j}$. Each $A_{j}$ is non-empty. Let $\{F_{1},\dots,F_{m}\}$ be a maximal set of pairwise disjoint connected subgraphs constructed as follows. The subgraph $F_{1}$ is induced by a minimal BFS subtree $S_{1}$ in $C$ rooted at some vertex $v\in V(C)$ and with $S_{1}$ intersecting all of $A_{1},\dots,A_{k}$. For $j\geqslant 1$, $F_{j+1}$ is induced by a minimal BFS subtree $S_{j+1}$ in C-\bigl{(}V(F_{1})\cup\dots\cup V(F_{j})\bigr{)} rooted at some vertex $v$ that is adjacent to $F_{1}\cup\dots\cup F_{j}$, and with $S_{j+1}$ intersecting all of $A_{1},\dots,A_{k}$. By minimality, each $S_{j}$ has at most $k\leqslant s$ leaves. Thus each $S_{j}$ is the union of at most $s$ shortest paths in C-\bigl{(}V(F_{1})\cup\dots\cup V(F_{j-1})\bigr{)}.

Suppose that $k\leqslant s-1$. Let $H_{i+1}=F_{1}$. Then $H_{i+1}$ satisfies the claim. Consider a component $C^{\prime}$ of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)}. If $C^{\prime}$ is disjoint from $C$, then $C^{\prime}$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} and $C^{\prime}$ is not adjacent to $H_{i+1}$. Otherwise, $C^{\prime}$ is contained in $C$, and the subgraphs in $H_{1},\dots,H_{i+1}$ that are adjacent to $C^{\prime}$ are a subset of the at most $s$ subgraphs $Q_{1},\dots,Q_{k},H_{i+1}$, which are pairwise adjacent since $F_{1}$ intersects all of $A_{1},\dots,A_{k}$. In both cases, property $(\star)$ is maintained.

Now assume that $k=s$. If $m\geqslant t$, then contracting each of $Q_{1},\dots,Q_{s},F_{1},\dots,F_{t}$ to a single vertex gives a $K^{*}_{s,t}$ minor. So we are left with the case $m\leqslant t-1$. Let $H_{i+1}$ be the subgraph of $C$ induced by $V(F_{1})\cup\dots\cup V(F_{m})$. Hence $H_{i+1}$ is induced by the union of $p_{i+1}$ paths $P_{1},\dots,P_{p_{i+1}}$, where $p_{i+1}\leqslant ms\leqslant s(t-1)$ and each $P_{j}$ is a shortest path in G-\Bigl{(}\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)}\cup\bigl{(}V(P_{1})\cup\dots\cup V(P_{j-1})\bigr{)}\Bigr{)}. Consider a component $C^{\prime}$ of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i+1})\bigr{)}. If $C^{\prime}$ is disjoint from $C$, then $C^{\prime}$ is a component of G-\bigl{(}V(H_{1})\cup\dots\cup V(H_{i})\bigr{)} and $C^{\prime}$ is not adjacent to $H_{i+1}$. Otherwise, $C^{\prime}$ is contained in $C$, and $C^{\prime}$ does not intersect some $A_{j}$ by the maximality of $m$. Thus $C^{\prime}$ is adjacent to a subset of at most $s$ subgraphs in $Q_{1},\dots,Q_{s},H_{i+1}$, which are pairwise adjacent (since $H_{i+1}$ intersects all of $A_{1},\dots,A_{s}$). In both cases, property $(\star)$ is maintained. ∎

5.1 Layered Treewidth and Generalised Colouring Numbers

This subsection explores connections between layered treewidth and strong colouring numbers. The layered width of a tree decomposition $(T_{x}:x\in V(T))$ of a graph $G$ is the minimum integer $k$ such that, for some layering $(V_{1},\dots,V_{\ell})$ of $G$, each bag $T_{x}$ contains at most $k$ vertices in each layer $V_{i}$. The layered treewidth of a graph $G$ is the minimum layered width of a tree decomposition of $G$. Layered treewidth was introduced independently by Dujmović et al. [18] and Shahrokhi [54]. Applications of layered treewidth include nonrepetitive graph colouring [18], queue and track layouts [18], graph drawing [18, 4], book embeddings [17], and intersection graph theory [54].

Lemma 30**.**

Every graph $G$ with layered treewidth $k$ satisfies $\operatorname{scol}_{r}(G)\leqslant k(2r+1)$.

Proof.

Let $(T_{x}:x\in V(T))$ be a tree decomposition of $G$ with layered width $k$ with respect to some layering $(V_{1},\dots,V_{\ell})$ of $G$. Root $T$ at an arbitrary node $r$. For each vertex $v$ of $G$, let $h(v)$ be the node $x$ of $T$ closest to $r$ in $T$ such that $v\in T_{x}$, let $S_{v}$ be the subtree of $T$ rooted at $h(v)$, and let $X_{v}=V(S_{v})$.

Let $\preccurlyeq^{-}$ be the partial ordering of $V(G)$ such that if $X_{w}\subsetneq X_{v}$, then $v\preccurlyeq^{-}w$. Let $\preccurlyeq$ be any linear ordering of $V(G)$ that is an extension of $\preccurlyeq^{-}$. By the definition of a tree decomposition, for any edge $vw$ with $v\preccurlyeq w$ we have $X_{w}\subseteq X_{v}$ and both $v$ and $w$ are in $T_{h(w)}$ (since there must be some bag $T_{u}$ with $v,w\in V(T_{u})$). This also means that if $u_{1},\ldots,u_{k}$ is a path and $u_{i}$ is minimal among $u_{1},\ldots,u_{k}$ with respect to $\preccurlyeq$, then $X_{u_{j}}\subseteq X_{u_{i}}$ for all $j\in[k]$.

Now assume that $x\in S(G,\preccurlyeq,v)$ for some $v,x\in V(G)$. Thus $G$ contains a path $v,y_{1},\dots,y_{k},x$ of length at most $r$ with $x\preccurlyeq v\preccurlyeq y_{i}$ for all $i\in[k]$. By the above observations this means that $x\in V(T_{h(y_{k})})$ (since $x\preccurlyeq y_{k}$ and $y_{k}x$ is an edge), $X_{v}\subseteq X_{x}$ (since $x$ is minimal with respect to $\preccurlyeq$ on the path $v,y_{1},\dots,y_{k},x$), and $X_{y_{k}}\subseteq X_{v}$ (since $v$ is minimal with respect to $\preccurlyeq$ on the path $v,y_{1},\dots,y_{k}$). So we have $X_{y_{k}}\subseteq X_{v}\subseteq X_{x}$, hence $T_{h(v)}$ is on the path from $T_{h(x)}$ to $T_{h(y_{k})}$ in $T$. Since $x$ is in both $T_{h(x)}$ and $T_{h(y_{k})}$, $x$ must also be in every bag in the path from $T_{h(x)}$ to $T_{h(y_{k})}$ in $T$, and hence $x\in V(T_{h(v)})$. Moreover, if $v\in V_{i}$, then, since $x$ has distance at most $r$ from $v$ in $G$, we have $x\in V_{i-r}\cup V_{i-r+1}\cup\dots\cup V_{i+r}$. Since $|V(T_{h(v)})\cap V_{j}|\leqslant k$, there are at most $(2r+1)k$ such vertices $x$. It follows that $\operatorname{scol}_{r}(G)\leqslant k(2r+1)$. ∎

While $n$-vertex planar graphs may have treewidth as large as $\sqrt{n}$, Dujmović et al. [18] proved that every graph with Euler genus $g$ has layered treewidth at most $2g+3$. (More generally, Dujmović et al. [18] proved that a minor-closed class of graphs has bounded layered treewidth if and only if it excludes some apex graph as a minor.) Then Lemma 30 implies that every planar graph $G$ satisfies

[TABLE]

This result is close to the best known result, which is $\operatorname{scol}_{r}(G)\leqslant 5r+1$, proved by Van den Heuvel et al. [34]. More generally, by Lemma 30, every graph $G$ with Euler genus $g$ satisfies

[TABLE]

Again, this result is close to the best known result, which is $\operatorname{scol}_{r}(G)\leqslant(4g+5)r+2g+1$, again due to Van den Heuvel et al. [34].

Lemma 30 is also interesting because it leads to linear bounds on $\operatorname{scol}_{r}$ for non-minor-closed classes. We give three examples.

A graph is $(g,k)$-planar if it can be drawn on a surface of Euler genus at most $g$ with at most $k$ crossings on each edge. Even $(0,1)$-planar graphs can contain arbitrarily large complete graph minors [16]. Nevertheless, Dujmović et al. [16] proved that every $(g,k)$-planar graph has layered treewidth at most $(4g+6)(k+1)$, and this bound is tight up to a constant factor. Then Lemma 30 implies that for every $(g,k)$-planar graph $G$,

[TABLE]

Map graphs are defined as follows. Start with a graph $G_{0}$ embedded in a surface of Euler genus $g$, with each face labelled a ‘nation’ or a ‘lake’, such that each vertex of $G_{0}$ is incident with at most $d$ nations. Define a graph $G$ whose vertices are the nations of $G_{0}$, where two vertices are adjacent in $G$ if the corresponding faces in $G_{0}$ share a vertex. Then $G$ is called a $(g,d)$-map graph. A $(0,d)$-map graph is called a (plane) $d$-map graph; see [6, 11, 7, 16] for example. It is easily seen that $(g,3)$-map graphs are precisely the graphs of Euler genus at most $g$ [7, 16]. So $(g,d)$-map graphs provide a natural generalisation of graphs embedded in a surface. Note that if a vertex of $G_{0}$ is incident with $d$ nations, then $G$ contains $K_{d}$, so map graphs can have arbitrarily large complete minors. Dujmović et al. [16] proved that every $(g,d)$-map graph on $n$ vertices has layered treewidth at most $(2g+3)(2d+1)$, and this bound is tight up to a constant factor. So Lemma 30 implies that for every $(g,d)$-map graph $G$,

[TABLE]

For a set $P$ of points in the plane, the unit disc graph $G$ of $P$ has vertex set $P$, where $vw\in E(G)$ if and only if $\operatorname{dist}(v,w)\leqslant 1$ (where now $\operatorname{dist}(v,w)$ denotes the Eulerian distance in the plane). Bannister et al. [4] proved that every unit disc graph with maximum clique size $k$ has layered pathwidth, and thus layered treewidth, at most $4k$. Then Lemma 30 implies that every unit disc graph with maximum clique size $k$ satisfies

[TABLE]

6 Excluded Immersions

This section studies the defective chromatic number of graphs excluding a fixed immersion. A graph $G$ contains a graph $H$ as an immersion (also called a weak immersion) if the vertices of $H$ can be mapped to distinct vertices of $G$, and the edges of $H$ can be mapped to pairwise edge-disjoint paths in $G$, such that each edge $vw$ of $H$ is mapped to a path in $G$ whose endpoints are the images of $v$ and $w$. The image in $G$ of each vertex in $H$ is called a branch vertex. A graph $G$ contains a graph $H$ as a strong immersion if $G$ contains $H$ as an immersion such that for each edge $vw$ of $H$, no internal vertex of the path in $G$ corresponding to $vw$ is a branch vertex.

Inspired no doubt by Hadwiger’s Conjecture, Lescure and Meyniel [44] and Abu-Khzam and Langston [1] independently conjectured that every $K_{t}$-immersion-free graph is properly $(t-1)$-colourable. Often motivated by this question, structural and colouring properties of graphs excluding a fixed immersion have recently been widely studied [27, 59, 22, 13, 20, 14, 51]. The best upper bound, due to Gauthier et al. [26], says that every $K_{t}$-immersion-free graph is properly $(3.54t+3)$-colourable.

We prove that the defective chromatic number of $K_{t}$-immersion-free graphs equals 2.

Theorem 31**.**

Every graph not containing $K_{t}$ as an immersion is $2$-colourable with defect $(t-1)^{3}$.

Theorem 32**.**

For every integer $t$, there is an integer $d$ such that every graph not containing $K_{t}$ as a strong immersion is $2$-colourable with defect $d$.

Notice that immersions naturally also appear in the setting of multigraphs, allowing multiple edges but no loops. It is obvious that if $G$ is a multigraph with edge multiplicity at most $m$, then the results of the theorems above hold with defect $m(t-1)^{3}$ and $md$, respectively. On the other hand, if every edge in a multigraph has multiplicity $m+1$, then no two adjacent vertices get the same colour in a colouring with defect $m$. In particular, the graph obtained by replacing the edges in the complete graph $K_{t-1}$ by $m+1$ parallel edges does not have $K_{t}$ as an immersion, but is also not $(t-2)$-colourable with defect $m$.

We leave as an open problem to determine the clustered chromatic number of graphs excluding a (strong or weak) $K_{t}$ immersion. It was observed by both Haxell et al. [33] and Liu and Oum [45] that the results in Alon et al. [2] prove that for every $k,N$, there exists a $(4k-2)$-regular graph $G$ such that every $k$-colouring of $G$ has a monochromatic component of size at least $N$. In other words, the clustered chromatic number of graphs with maximum degree $\Delta$ is at least \bigl{\lfloor}\frac{1}{4}(\Delta+6)\bigr{\rfloor}. Since every graph with maximum degree at most $t-2$ contains no (strong or weak) $K_{t}$ immersion, the clustered chromatic number of graphs excluding a (strong or weak) $K_{t}$ immersion is at least \bigl{\lfloor}\frac{1}{4}(t+4)\bigr{\rfloor}.

The proof of Theorem 31 uses the following structure theorem from DeVos et al. [14]. The theorem is not explicitly proved in [14], but can be derived easily from the proof of Theorem 1 on page 4 of that paper.

For each edge $xy$ of a tree $T$, let $T(xy)$ and $T(yx)$ be the components of $T-xy$, where $x$ is in $T(xy)$ and $y$ is in $T(yx)$. For a tree $T$ and graph $G$, a $T$-partition of $G$ is a partition \bigl{(}T_{x}\subseteq V(G):x\in V(T)\bigr{)} of $V(G)$ indexed by the nodes of $T$. As before, each set $T_{x}$ is called a bag. Note that a bag may be empty. For each edge $xy\in E(T)$, let $G(T,xy)=\bigcup_{z\in V(T(xy))}T_{z}$ and $G(T,yx)=\bigcup_{z\in V(T(yx))}T_{z}$. Let $E(T,xy)$ ($=E(T,yx)$) be the set of edges in $G$ between $G(T,xy)$ and $G(T,yx)$. The adhesion of a $T$-partition is the maximum, taken over all edges $xy$ of $T$, of $|E(T,xy)|$. For each node $x$ of $T$, the torso of $x$ (with respect to a $T$-partition) is the graph obtained from $G$ by identifying $G(T,yx)$ into a single vertex for each edge $xy$ incident to $x$, deleting resulting parallel edges and loops.

Theorem 33** (following DeVos et al. [14]).**

For every graph $H$ with $t$ vertices and every graph $G$ that does not contain $H$ as an immersion, there is a tree $T$ and a $T$-partition of $\,G$ with adhesion less than $(t-1)^{2}$, such that each bag has at most $t-1$ vertices.

A structural result similar to this theorem was proved by Wollan [59]. We also need the following lemma.

Lemma 34**.**

Let $G$ be a graph such that for some tree $T$ with vertex set $V(G)$, for each edge $xy$ of $\,T$, the number of edges of $\,G$ between $V(T(xy))$ and $V(T(yx))$ is at most $k$. Then $G$ is $2$-colourable with defect $k$.

Proof.

We use induction on $|V(G)|$, noting that there is nothing to prove if $|V(G)|\leqslant 2$. So assume $|V(G)|\geqslant 3$. Call a vertex $v$ of $G$ large if $\deg_{G}(v)\geqslant k+1$; otherwise $v$ is small.

If $G$ has no large vertices, then every 2-colouring of $G$ has defect $k$. Now assume that $G$ has some large vertex. Thus there is an edge $uv$ of $T$ such that $u$ is large and $u$ is the only large vertex in $V(T(uv))$. Set $a=|V(T(uv))|$. Suppose that every vertex in $V(T(uv))\setminus\{u\}$ has a neighbour in $G$ in $V(T(vu))$. Since $u$ has at least $k+1-(a-1)$ neighbours outside $V(T(uv))$, the number of edges between $V(T(uv))$ and $V(T(vu))$ is at least \bigl{(}k+1-(a-1)\bigr{)}+(a-1)=k+1, a contradiction.

So there is a vertex $w\in V(T(uv))\setminus\{u\}$ with $N_{G}(w)\subseteq V(T(uv))$. Note that $w$ is small. Let $wz$ be an edge in $T$. Form the graphs $G^{\prime}$ and $T^{\prime}$ respectively from $G$ and $T$ by identifying $w$ and $z$ (deleting loops and parallel edges). For each edge $xy$ of $T^{\prime}$, the number of edges of $G^{\prime}$ between $V(T^{\prime}(xy))$ and $V(T^{\prime}(yx))$ is still at most $k$. Hence by induction, $G^{\prime}$ has a $2$-colouring with defect $k$. This colouring gives a $2$-colouring with defect $k$ of all vertices of $G$ except $w$. Since all vertices in $V(T(uv))$ except $u$ are small, $u$ is the only possible large neighbour of $w$. Give $w$ the colour different from $u$. As all other neighbours of $w$ are small, the monochromatic degree can increase only for small vertices. Thus the defect is at most $k$, as required. ∎

Now we are ready to prove our $2$-colour result for graphs excluding an immersion.

Proof of Theorem 31..

By Theorem 33, there is a tree $T$ and a $T$-partition of $G$ with adhesion at most $(t-1)^{2}-1$, such that each bag has at most $t-1$ vertices. Let $Q$ be the graph with vertex set $V(T)$, where $xy\in E(Q)$ whenever there is an edge of $G$ between $T_{x}$ and $T_{y}$. Any one edge of $Q$ corresponds to at most $t-1$ edges in $G$. By Lemma 34, the graph $Q$ is $2$-colourable with defect $(t-1)^{2}-1$. Assign to each vertex $v$ in $G$ the colour assigned to the vertex $x$ in $Q$ with $v\in T_{x}$. Since at most $t-1$ vertices of $G$ are in each bag, $G$ is $2$-coloured with defect at most (t-1)\cdot\bigl{(}(t-1)^{2}-1\bigr{)}+(t-2)<(t-1)^{3}. ∎

To prove our result for strong immersions, we employ the following more involved structure theorem of Dvořák and Wollan [22].

Theorem 35** (Dvořák and Wollan [22]).**

For every integer $t$, there is an integer $\alpha$ such that for every graph $G$ that does not contain $K_{t}$ as a strong immersion, there is a tree $T$ and a $T$-partition of $\,G$ with adhesion at most $\alpha$ such that the following holds. For each node $x$ of $\,T$ with torso $S_{x}$, if $W_{x}$ is the set of vertices in $S_{x}$ with degree at least $\alpha$, then there is a subset $A_{x}\subseteq W_{x}$ of size at most $\alpha$ such that $W_{x}\setminus A_{x}$ can be enumerated $\{x_{1},\dots,x_{p}\}$ and $V(S_{x}-W_{x})$ can be partitioned $B_{0},B_{1},\dots,B_{p}$ (allowing $B_{j}=\varnothing$), such that:

1. (1)

each vertex $v\in A_{x}$ is adjacent to at most $\alpha$ of $\,B_{0},B_{1},\dots,B_{p}$ and adjacent to at most $\alpha$ vertices in $W_{x}\setminus A_{x}$; and 2. (2)

for each $i\in[p]$, there are at most $\alpha$ edges between $B_{0}\cup\dots\cup B_{i-1}\cup\{x_{1},\dots,x_{i-1}\}$ and $B_{i}\cup\dots\cup B_{p}\cup\{x_{i+1},\dots,x_{p}\}$.

We actually only need the following corollary of Theorem 35.

Corollary 36**.**

For every integer $t$, there is an integer $\alpha$ such that for every graph $G$ that does not contain $K_{t}$ as a strong immersion, there is a tree $T$ and $T$-partition of $\,G$ with adhesion at most $\alpha^{2}$ such that for each node $x$ of $\,T$ with torso $S_{x}$, $S_{x}[T_{x}]$ has degree at most $3\alpha+2$.

Proof.

Consider a tree $T$ and $T$-partition of $G$ in accordance with Theorem 35. Consider a node $x$ of $T$ with torso $S_{x}$. We use the notation from the theorem.

Consider a vertex $v\in W_{x}$. If $v\in A_{x}$, then $v$ has at most $|A_{x}|-1<\alpha$ neighbours in $A_{x}$ and at most $\alpha$ neighbours in $W_{x}\setminus A_{x}$, and thus has less than $2\alpha$ neighbours in $W_{x}$. If $v\in W_{x}\setminus A_{x}$, then $v=x_{i}$ for some $i\in[p]$. Then $v$ has at most $|A_{x}|\leqslant\alpha$ neighbours in $A_{x}$. Furthermore, there are at most $\alpha$ edges between $\{x_{1},\dots,x_{i-2}\}$ and $\{x_{i},\dots,x_{p}\}$, at most $\alpha$ edges between $\{x_{1},\dots,x_{i}\}$ and $\{x_{i+2},\dots,x_{p}\}$, and at most $2$ edges between $x_{i}$ and $\{x_{i-1},x_{i+1}\}$. Thus $v$ has at most $3\alpha+2$ neighbours in $W_{x}$. Hence $S_{x}[W_{x}]$ has maximum degree at most $3\alpha+2$.

Apply the following operation for each vertex $v\in T_{x}\setminus W_{x}$, for each node $x$ of $T$ with $|T_{x}|\geqslant 2$. Since $v\not\in W_{x}$, the degree of $v$ in $S_{x}$ is at most $\alpha-1$. Since there are at most $\alpha$ edges from $G$ between $T_{x}$ and each contracted vertex in $S_{x}$, $v$ has degree at most $(\alpha-1)\alpha<\alpha^{2}$ in $G$. Now delete $v$ from $T_{x}$, add a new node $y$ in $T$ adjacent only to $x$, and define $T_{y}=\{v\}$. Note that the number of edges between $T_{y}$ and $G-T_{y}$ is less than $\alpha^{2}$, and the torso of $y$ is isomorphic to $K_{2}$ (hence has degree one). Finally, the torso of $x$ hasn’t changed, since the contraction of the single-vertex node $T_{y}$ just gives the vertex $v$ again. In particular, the degree of $v$ in the torso of $x$ is still at most $\alpha-1$, and hence $W_{x}$ also hasn’t changed.

After having applied the operation from the previous paragraph as long a possible, we obtain a tree-partition of $G$ on a tree $T^{\prime}$ with adhesion at most $\alpha^{2}$. Moreover, for each node $x$ of $T^{\prime}$ we have $|T^{\prime}_{x}|=1$, and then $S^{\prime}_{x}[T^{\prime}_{x}]$ has degree zero, or $T^{\prime}_{x}\subseteq W^{\prime}_{x}$ and $S^{\prime}_{x}[W^{\prime}_{x}]$ has degree at most $3\alpha+2$. We immediately get that $S^{\prime}_{x}[T^{\prime}_{x}]$ has degree at most $3\alpha+2$ as well. ∎

Now we are ready to prove our $2$-colouring result for graphs excluding a strong immersion.

Proof of Theorem 32..

By 36, there is an integer $\alpha$, a tree $T$ and a $T$-partition of $G$ with adhesion at most $\alpha^{2}$, such that for each node $x$ of $T$ with torso $S_{x}$, $S_{x}[T_{x}]$ has degree at most $3\alpha+2$. Let $Q$ be the graph with vertex set $V(T)$, where $xy\in E(Q)$ whenever there is an edge of $G$ between $T_{x}$ and $T_{y}$. By Lemma 34 with $k=\alpha^{2}$, the graph $Q$ is $2$-colourable with defect $\alpha^{2}$. Assign to each vertex $v$ in $G$ the colour assigned to the vertex $x$ in $Q$ with $v\in T_{x}$.

If $v\in V(T_{x})$, then every edge $vw$ in $G$ with $w\not\in T_{x}$ gives rise to an edge in $Q$. Since the adhesion is at most $\alpha^{2}$, any one edge of $Q$ corresponds to at most $\alpha^{2}$ edges in $G$. As the monochromatic degree of $x$ in $Q$ is at most $\alpha^{2}$, this means that $v$ has at most $\alpha^{4}$ neighbours outside $T_{x}$ with the same colour. Adding the at most $3\alpha+2$ neighbours of $v$ in $T_{x}$, we obtain that the monochromatic degree of $v$ in $G$ is at most $\alpha^{4}+3\alpha+2$. ∎

Notes and acknowledgements

After publication of the first version of this paper, Dvořák and Norin [21] proved that every graph embeddable in a fixed surface is $4$-colourable with bounded clustering (cf. the comments after Theorem 17). They also gave an alternative proof that the clustered chromatic number of $K_{t}$-minor-free graphs is at most $2t-2$ (cf. Theorem 1), and announced that in a sequel they will prove that the clustered chromatic number of $K_{t}$-minor-free graphs equals $t-1$.

Thanks to Jacob Fox who first observed that bounded degree graphs give lower bounds on the clustered chromatic number of graphs excluding a fixed immersion.

The authors also like to thank an anonymous referee for pointing out some errors in earlier versions of this paper.