DP-colorings of graphs with high chromatic number

TL;DR

This paper proves that for large graphs with chromatic number nearly equal to the number of vertices, the DP-chromatic number matches the chromatic number, highlighting a key difference from list coloring.

Contribution

It establishes a tight bound showing DP-coloring equals chromatic number for graphs with high chromatic number, extending understanding of DP-coloring's behavior.

Findings

DP-chromatic number equals chromatic number for graphs with hi(G) e n - O( n)

Lower bound on hi_{DP}(G) is tight up to constant factors

Contrast with list coloring where the bound is not tight

Abstract

DP-coloring is a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle. We prove that for every $n$-vertex graph $G$ whose chromatic number $\chi(G)$ is "close" to $n$, the DP-chromatic number of $G$ equals $\chi(G)$. "Close" here means $\chi(G)\geq n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$), in contrast to the case of list coloring.

Equations

$${\chi }_{\ell }(\mathbf{J}(G,s))=\chi (\mathbf{J}(G,s)),$$

$$Z_{\ell }(G):=\min \{s\in \mathbb{N}:{\chi }_{\ell }(\mathbf{J}(G,s))=\chi (\mathbf{J}(G,s))\},$$

$$Z_{\ell }(n):=\max \{Z_{\ell }(G):|V(G)|=n\}.$$

$$(u_1,c_1)(u_2,c_2)\in E(H):⟺(u_1=u_2\text{ and }{c}_1\ne c_2)\text{ or }(u_1{u}_2\in E(G)\text{ and }{c}_1=c_2).$$

$$Z_{DP}(G):=\min \{s\in \mathbb{N}:{\chi }_{DP}(\mathbf{J}(G,s))=\chi (\mathbf{J}(G,s))\},$$

$$Z_{DP}(n):=\max \{Z_{DP}(G):|V(G)|=n\}.$$

$$Z_{DP}(G)\le 3m-\frac{3}{2}(k-1)n.$$

$$n-r(n)=\Theta (\sqrt{n}).$$

$$|A|\ge \frac{3}{2}\sum _{v\in V(G)}\max \{{\mathrm{deg}}_G(v)+|A|-|L(v)|+1,0\}.$$

$$\sigma (G,A,L,H,v):=\max \{{\mathrm{deg}}_G(v)+|A|-|L(v)|+1,0\}$$

$$\sigma (G,A,L,H):=\sum _{v\in V(G)}\sigma (G,A,L,H,v).$$

$$\sigma (v):=\sigma (G,A,L,H,v);\sigma :=\sigma (G,A,L,H).$$

$$|A|\ge \frac{3\sigma }{2}.$$

$$\frac{3}{2}\sigma (G^{\prime },A,L,H)\le \frac{3\sigma }{2}\le |A|.$$

$$\sigma (v)={\mathrm{deg}}_G(v)+|A|-|L(v)|+1>0.$$

$${\mathrm{deg}}_G(v_0)+|A|-|L(v_0)|+1\le 0,$$

$$|L(v_0)|\ge {\mathrm{deg}}_G(v_0)+|A|+1.$$

$$|L(v_0)|-({\mathrm{deg}}_G(v_0)+|A|)\ge 1$$

$$\frac{3}{2}\sigma (G,A^{\prime },L^{\prime },H)\le \frac{3(\sigma -1)}{2}\le |A|-\frac{3}{2}<|A^{\prime }|.$$

$$|L(v)|\ge |L(a)|\ge |A|+k.$$

$$0\le {\mathrm{deg}}_G(v)+|A|-|L(v)|\le {\mathrm{deg}}_G(v)-k,$$

$$\frac{3}{2}\sigma (G,A^{\prime },L^{\prime },H)\le \frac{3(\sigma -2)}{2}\le |A|-3\le |A^{\prime }|.$$

$$I_0:=\{y(u):u\in U_0\}\text{ and }{I}_1:=\{y(u):u\in U_1\}.$$

$$|L^{\prime }(a_1)|=|L(a_1)|-1\ge |A|+(k-1).$$

$$|L^{\prime }(a)|=|L(a)\setminus N_H(I_0)|\ge |L(a)\setminus N_H(x_1)|\ge |L(a)|-1\ge |A|+(k-1).$$

$$x_0\in (L(a_0)\cap N_H(I_0))\setminus N_H(x_1).$$

$$|L^{\prime }(a_1)|\ge |L(a_1)|-2\ge |A|+k-2=|A^{\prime }|+(k-1).$$

$$|L^{\prime }(a)|\ge |L(a)\setminus N_H(x_0,x_1)|\ge |L(a)|-2\ge |A|+k-2=|A^{\prime }|+(k-1).$$

$$x_2\in (L(a_2)\cap N_H(I_1))\setminus N_H(x_0,x_1).$$

$$\frac{3}{2}\sum _{v\in V(G)}\max \{{\mathrm{deg}}_G(v)-|L(v)|+|A|+1,0\}=\frac{3}{2}\sum _{v\in V(G)}\max \{{\mathrm{deg}}_G(v)-k+1,0\}.$$

Full text

1 Introduction

We use standard notation. In particular, $\mathbb{N}$ denotes the set of all nonnegative integers. For a set $S$, $\operatorname{Pow}(S)$ denotes the power set of $S$, i.e., the set of all subsets of $S$. All graphs considered here are finite, undirected, and simple. For a graph $G$, $V(G)$ and $E(G)$ denote the vertex and the edge sets of $G$, respectively. For a set $U\subseteq V(G)$, $G[U]$ is the subgraph of $G$ induced by $U$. Let $G-U\coloneqq G[V(G)\setminus U]$, and for $u\in V(G)$, let $G-u\coloneqq G-\{u\}$. For $U_{1}$, $U_{2}\subseteq V(G)$, let $E_{G}(U_{1},U_{2})\subseteq E(G)$ denote the set of all edges in $G$ with one endpoint in $U_{1}$ and the other one in $U_{2}$. For $u\in V(G)$, $N_{G}(u)\subset V(G)$ denotes the set of all neighbors of $u$, and $\deg_{G}(u)\coloneqq|N_{G}(u)|$ is the degree of $u$ in $G$. We use $\delta(G)$ to denote the minimum degree of $G$, i.e., $\delta(G)\coloneqq\min_{u\in V(G)}\deg_{G}(u)$. For $U\subseteq V(G)$, let $N_{G}(U)\coloneqq\bigcup_{u\in U}N_{G}(u)$. To simplify notation, we write $N_{G}(u_{1},\ldots,u_{k})$ instead of $N_{G}(\{u_{1},\ldots,u_{k}\})$. A set $I\subseteq V(G)$ is independent if $I\cap N_{G}(I)=\varnothing$, i.e., if $uv\not\in E(G)$ for all $u$, $v\in I$. We denote the family of all independent sets in a graph $G$ by $\mathcal{I}(G)$. The complete $k$-vertex graph is denoted by $K_{k}$.

1.1 The Noel–Reed–Wu Theorem for list coloring

Recall that a proper coloring of a graph $G$ is a function $f\colon V(G)\to Y$, where $Y$ is a set of colors, such that $f(u)\neq f(v)$ for every edge $uv\in E(G)$. The smallest $k\in\mathbb{N}$ such that there exists a proper coloring $f\colon V(G)\to Y$ with $|Y|=k$ is called the chromatic number of $G$ and is denoted by $\chi(G)$.

List coloring was introduced independently by Vizing [11] and Erdős, Rubin, and Taylor [6]. A list assignment for a graph $G$ is a function $L\colon V(G)\to\operatorname{Pow}(Y)$, where $Y$ is a set. For each $u\in V(G)$, the set $L(u)$ is called the list of $u$, and its elements are the colors available for $u$. A proper coloring $f\colon V(G)\to Y$ is called an $L$-coloring if $f(u)\in L(u)$ for each $u\in V(G)$. The list chromatic number $\chi_{\ell}(G)$ of $G$ is the smallest $k\in\mathbb{N}$ such that $G$ is $L$-colorable for each list assignment $L$ with $|L(u)|\geq k$ for all $u\in V(G)$. It is an immediate consequence of the definition that $\chi_{\ell}(G)\geq\chi(G)$ for every graph $G$.

It is well-known (see, e.g., [6, 11]) that the list chromatic number of a graph can significantly exceed its ordinary chromatic number. Moreover, there exist $2$-colorable graphs with arbitrarily large list chromatic numbers. On the other hand, Noel, Reed, and Wu [7] established the following result, which was conjectured by Ohba [8, Conjecture 1.3]:

Theorem 1.1** (Noel–Reed–Wu [7]).**

Let $G$ be an $n$-vertex graph with $\chi(G)\geq(n-1)/2$. Then $\chi_{\ell}(G)=\chi(G)$.

The following construction was first studied by Ohba [8] and Enomoto, Ohba, Ota, and Sakamoto [5]. For a graph $G$ and $s\in\mathbb{N}$, let $\mathbf{J}(G,s)$ denote the join of $G$ and a copy of $K_{s}$, i.e., the graph obtained from $G$ by adding $s$ new vertices that are adjacent to every vertex in $V(G)$ and to each other. It is clear from the definition that for all $G$ and $s$, $\chi(\mathbf{J}(G,s))=\chi(G)+s$. Moreover, we have $\chi_{\ell}(\mathbf{J}(G,s))\leq\chi_{\ell}(G)+s$; however, this inequality can be strict. Indeed, Theorem 1.1 implies that for every graph $G$ and every $s\geq|V(G)|-2\chi(G)-1$,

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even if $\chi_{\ell}(G)$ is much larger than $\chi(G)$. In view of this observation, it is interesting to consider the following parameter:

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i.e., the smallest $s\in\mathbb{N}$ such that the list and the ordinary chromatic numbers of $\mathbf{J}(G,s)$ coincide. The parameter $Z_{\ell}(G)$ was explicitly defined by Enomoto, Ohba, Ota, and Sakamoto in [5, page 65] (they denoted it $\psi(G)$). Recently, Kim, Park, and Zhu (personal communication, 2016) obtained new lower bounds on $Z_{\ell}(K_{2,n})$, $Z_{\ell}(K_{n,n})$, and $Z_{\ell}(K_{n,n,n})$. One can also consider, for $n\in\mathbb{N}$,

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The parameter $Z_{\ell}(n)$ is closely related to the Noel–Reed–Wu Theorem, since, by definition, there exists a graph $G$ on $n+Z_{\ell}(n)-1$ vertices whose ordinary chromatic number is at least $Z_{\ell}(n)$ and whose list and ordinary chromatic numbers are distinct. The finiteness of $Z_{\ell}(n)$ for all $n\in\mathbb{N}$ was first established by Ohba [8, Theorem 1.3]. Theorem 1.1 yields an upper bound $Z_{\ell}(n)\leq n-5$ for all $n\geq 5$; on the other hand, a result of Enomoto, Ohba, Ota, and Sakamoto [5, Proposition 6] implies that $Z_{\ell}(n)\geq n-O(\sqrt{n})$.

1.2 DP-colorings and the results of this paper

The goal of this note is to study analogs of $Z_{\ell}(G)$ and $Z_{\ell}(n)$ for the generalization of list coloring that was recently introduced by Dvořák and Postle [4], which we call DP-coloring. Dvořák and Postle invented DP-coloring to attack an open problem on list coloring of planar graphs with no cycles of certain lengths.

Definition 1.2**.**

Let $G$ be a graph. A cover of $G$ is a pair $(L,H)$, where $H$ is a graph and $L\colon V(G)\to\operatorname{Pow}(V(H))$ is a function, with the following properties:

• –

the sets $L(u)$, $u\in V(G)$, form a partition of $V(H)$;

• –

if $u$, $v\in V(G)$ and $L(v)\cap N_{H}(L(u))\neq\varnothing$, then $v\in\{u\}\cup N_{G}(u)$;

• –

each of the graphs $H[L(u)]$, $u\in V(G)$, is complete;

• –

if $uv\in E(G)$, then $E_{H}(L(u),L(v))$ is a matching (not necessarily perfect and possibly empty).

Definition 1.3**.**

Let $G$ be a graph and let $(L,H)$ be a cover of $G$. An $(L,H)$-coloring of $G$ is an independent set $I\in\mathcal{I}(H)$ of size $|V(G)|$. Equivalently, $I\in\mathcal{I}(H)$ is an $(L,H)$-coloring of $G$ if $|I\cap L(u)|=1$ for all $u\in V(G)$.

Remark 1.4**.**

Suppose that $G$ is a graph, $(L,H)$ is a cover of $G$, and $G^{\prime}$ is a subgraph of $G$. In such situations, we will allow a slight abuse of terminology and speak of $(L,H)$-colorings of $G^{\prime}$ (even though, strictly speaking, $(L,H)$ is not a cover of $G^{\prime}$).

The DP-chromatic number $\chi_{DP}(G)$ of a graph $G$ is the smallest $k\in\mathbb{N}$ such that $G$ is $(L,H)$-colorable for each cover $(L,H)$ with $|L(u)|\geq k$ for all $u\in V(G)$.

To show that DP-colorings indeed generalize list colorings, consider a graph $G$ and a list assignment $L$ for $G$. Define a graph $H$ as follows: Let $V(H)\coloneqq\{(u,c)\,:\,u\in V(G)\text{ and }c\in L(u)\}$ and let

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For $u\in V(G)$, let $\hat{L}(u)\coloneqq\{(u,c)\,:\,c\in L(u)\}$. Then $(\hat{L},H)$ is a cover of $G$, and there is a one-to-one correspondence between $L$-colorings and $(\hat{L},H)$-colorings of $G$. Indeed, if $f$ is an $L$-coloring of $G$, then the set $I_{f}\coloneqq\{(u,f(u))\,:\,u\in V(G)\}$ is an $(\hat{L},H)$-coloring of $G$. Conversely, given an $(\hat{L},H)$-coloring $I$ of $G$, we can define an $L$-coloring $f_{I}$ of $G$ by the property $(u,f_{I}(u))\in I$ for all $u\in V(G)$. Thus, list colorings form a subclass of DP-colorings. In particular, $\chi_{DP}(G)\geq\chi_{\ell}(G)$ for each graph $G$.

Some upper bounds on list-chromatic numbers hold for DP-chromatic numbers as well. For example, $\chi_{DP}(G)\leq d+1$ for any $d$-degenerate graph $G$. Dvořák and Postle [4] pointed out that Thomassen’s bounds [9, 10] on the list chromatic numbers of planar graphs hold also for their DP-chromatic numbers; in particular, $\chi_{DP}(G)\leq 5$ for every planar graph $G$. On the other hand, there are also some striking differences between DP- and list coloring. For instance, even cycles are $2$-list-colorable, while their DP-chromatic number is $3$; in particular, the orientation theorems of Alon–Tarsi [2] and the Bondy–Boppana–Siegel Lemma (see [2]) do not extend to DP-coloring (see [3] for further examples of differences between list and DP-coloring).

By analogy with (1.1) and (1.2), we consider the parameters

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and

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Our main result asserts that for all graphs $G$, $Z_{DP}(G)$ is finite:

Theorem 1.5**.**

Let $G$ be a graph with $n$ vertices, $m$ edges, and chromatic number $k$. Then $Z_{DP}(G)\leq 3m$. Moreover, if $\delta(G)\geq k-1$, then

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Corollary 1.6**.**

For all $n\in\mathbb{N}$, $Z_{DP}(n)\leq 3n^{2}/2$.

Note that the upper bound on $Z_{DP}(n)$ given by Corollary 1.6 is quadratic in $n$, in contrast to the linear upper bound on $Z_{\ell}(n)$ implied by Theorem 1.1. Our second result shows that the order of magnitude of $Z_{DP}(n)$ is indeed quadratic:

Theorem 1.7**.**

For all $n\in\mathbb{N}$, $Z_{DP}(n)\geq n^{2}/4-O(n)$.

Corollary 1.6 and Theorem 1.7 also yield the following analog of Theorem 1.1 for DP-coloring:

Corollary 1.8**.**

For $n\in\mathbb{N}$, let $r(n)$ denote the minimum $r\in\mathbb{N}$ such that for every $n$-vertex graph $G$ with $\chi(G)\geq r$, we have $\chi_{DP}(G)=\chi(G)$. Then

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We prove Theorem 1.5 in Section 2 and Theorem 1.7 in Section 3. The derivation of Corollary 1.8 from Corollary 1.6 and Theorem 1.7 is straightforward; for completeness, we include it at the end of Section 3.

2 Proof of Theorem 1.5

For a graph $G$ and a finite set $A$ disjoint from $V(G)$, let $\mathbf{J}(G,A)$ denote the graph with vertex set $V(G)\cup A$ obtained from $G$ be adding all edges with at least one endpoint in $A$ (i.e., $\mathbf{J}(G,A)$ is a concrete representative of the isomorphism type of $\mathbf{J}(G,|A|)$).

First we prove the following more technical version of Theorem 1.5:

Theorem 2.1**.**

Let $G$ be a $k$-colorable graph. Let $A$ be a finite set disjoint from $V(G)$ and let $(L,H)$ be a cover of $\mathbf{J}(G,A)$ such that for all $a\in A$, $|L(a)|\geq|A|+k$. Suppose that

[TABLE]

Then $\mathbf{J}(G,A)$ is $(L,H)$-colorable.

Proof.

For a graph $G$, a set $A$ disjoint from $V(G)$, a cover $(L,H)$ of $\mathbf{J}(G,A)$, and a vertex $v\in V(G)$, let

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and

[TABLE]

Assume, towards a contradiction, that a tuple $(k,G,A,L,H)$ forms a counterexample which minimizes $k$, then $|V(G)|$, and then $|A|$. For brevity, we will use the following shortcuts:

[TABLE]

Thus, (2.1) is equivalent to

[TABLE]

Note that $|V(G)|$ and $|A|$ are both positive. Indeed, if $V(G)=\varnothing$, then $\mathbf{J}(G,A)$ is just a clique with vertex set $A$, so its DP-chromatic number is $|A|$. If, on the other hand, $A=\varnothing$, then (2.1) implies that $|L(v)|\geq\deg_{G}(v)+1$ for all $v\in V(G)$, so an $(L,H)$-coloring of $G$ can be constructed greedily. Furthermore, $\chi(G)=k$, since otherwise we could have used the same $(G,A,L,H)$ with a smaller value of $k$.

Claim 2.1.1**.**

For every $v\in V(G)$, the graph $\mathbf{J}(G-v,A)$ is $(L,H)$-colorable.

Proof.

Consider any $v_{0}\in V(G)$ and let $G^{\prime}\coloneqq G-v_{0}$. For all $v\in V(G^{\prime})$, $\deg_{G^{\prime}}(v)\leq\deg_{G}(v)$, and thus $\sigma(G^{\prime},A,L,H,v)\leq\sigma(v)$. Therefore,

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By the minimality of $|V(G)|$, the conclusion of Theorem 2.1 holds for $(k,G^{\prime},A,L,H)$, i.e., $\mathbf{J}(G^{\prime},A)$ is $(L,H)$-colorable, as claimed. ∎

Corollary 2.1.2**.**

For every $v\in V(G)$,

[TABLE]

Proof.

Suppose that for some $v_{0}\in V(G)$,

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i.e.,

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Using Claim 2.1.1, fix any $(L,H)$-coloring $I$ of $\mathbf{J}(G-v_{0},A)$. Since $v_{0}$ still has at least

[TABLE]

available colors, $I$ can be extended to an $(L,H)$-coloring of $\mathbf{J}(G,A)$ greedily; a contradiction. ∎

Claim 2.1.3**.**

For every $v\in V(G)$ and $x\in\bigcup_{a\in A}L(a)$, there is $y\in L(v)$ such that $xy\in E(H)$.

Proof.

Suppose that for some $a_{0}\in A$, $x_{0}\in L(a_{0})$, and $v_{0}\in V(G)$, we have $L(v_{0})\cap N_{H}(x_{0})=\varnothing$. Let $A^{\prime}\coloneqq A\setminus\{a_{0}\}$, and for every $w\in V(G)\cup A^{\prime}$, let $L^{\prime}(w)\coloneqq L(w)\setminus N_{H}(x_{0})$. Note that for all $a\in A^{\prime}$, $|L^{\prime}(a)|\geq|A^{\prime}|+k$, and for all $v\in V(G)$, $\sigma(G,A^{\prime},L^{\prime},H,v)\leq\sigma(v)$. Moreover, by the choice of $x_{0}$, $|L^{\prime}(v_{0})|=|L(v_{0})|$, which, due to Corollary 2.1.2, yields $\sigma(G,A^{\prime},L^{\prime},H,v_{0})\leq\sigma(v_{0})-1$. This implies $\sigma(G,A^{\prime},L^{\prime},H)\leq\sigma-1$, and thus

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By the minimality of $|A|$, the conclusion of Theorem 2.1 holds for $(k,G,A^{\prime},L^{\prime},H)$, i.e., the graph $\mathbf{J}(G,A^{\prime})$ is $(L^{\prime},H)$-colorable. By the definition of $L^{\prime}$, for any $(L^{\prime},H)$-coloring $I$ of $\mathbf{J}(G,A^{\prime})$, $I\cup\{x_{0}\}$ is an $(L,H)$-coloring of $\mathbf{J}(G,A)$. This is a contradiction. ∎

Corollary 2.1.4**.**

$k\geq 2$.

Proof.

Let $v\in V(G)$ and consider any $a\in A$. Since, by Claim 2.1.3, each $x\in L(a)$ has a neighbor in $L(v)$, we have

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Using Corollary 2.1.2, we obtain

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i.e., $\deg_{G}(v)\geq k$. Since $V(G)\neq\varnothing$, $k\geq 1$, which implies $\deg_{G}(v)\geq 1$. But then $\chi(G)\geq 2$, as desired. ∎

Claim 2.1.5**.**

$H$* does not contain a walk of the form $x_{0}-y_{0}-x_{1}-y_{1}-x_{2}$, where*

• •

$x_{0}$, $x_{1}$, $x_{2}\in\bigcup_{a\in A}L(a)$;

• •

$y_{0}$, $y_{1}\in\bigcup_{v\in V(G)}L(v)$;

• •

$x_{0}\neq x_{1}\neq x_{2}$* and $y_{0}\neq y_{1}$ (but it is possible that $x_{0}=x_{2}$);*

• •

the set $\{x_{0},x_{1},x_{2}\}$ is independent in $H$.

Proof.

Suppose that such a walk exists and let $a_{0}$, $a_{1}$, $a_{2}\in A$ and $v_{0}$, $v_{1}\in V(G)$ be such that $x_{0}\in L(a_{0})$, $y_{0}\in L(v_{0})$, $x_{1}\in L(a_{1})$, $y_{1}\in L(v_{1})$, and $x_{2}\in L(a_{2})$. Let $A^{\prime}\coloneqq A\setminus\{a_{0},a_{1},a_{2}\}$, and for every $w\in V(G)\cup A^{\prime}$, let $L^{\prime}(w)\coloneqq L(w)\setminus N_{H}(x_{0},x_{1},x_{2})$. Since $\{x_{0},x_{1},x_{2}\}$ is an independent set, for all $a\in A^{\prime}$, $|L^{\prime}(a)|\geq|A^{\prime}|+k$, while for all $v\in V(G)$, $\sigma(G,A^{\prime},L^{\prime},H,v)\leq\sigma(v)$. Moreover, since for each $i\in\{0,1\}$, the set $\{x_{0},x_{1},x_{2}\}$ contains two distinct neighbors of $y_{i}$, we have $\sigma(G,A^{\prime},L^{\prime},H,v_{i})\leq\sigma(v_{i})-1$. Therefore, $\sigma(G,A^{\prime},L^{\prime},H)\leq\sigma-2$, and thus

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By the minimality of $|A|$, the conclusion of Theorem 2.1 holds for $(k,G,A^{\prime},L^{\prime},H)$, i.e., the graph $\mathbf{J}(G,A^{\prime})$ is $(L^{\prime},H)$-colorable. By the definition of $L^{\prime}$, for any $(L^{\prime},H)$-coloring $I$ of $\mathbf{J}(G,A^{\prime})$, $I\cup\{x_{0},x_{1},x_{2}\}$ is an $(L,H)$-coloring of $\mathbf{J}(G,A)$. This is a contradiction. ∎

Due to Corollary 2.1.4, we can choose a pair of disjoint independent sets $U_{0}$, $U_{1}\subset V(G)$ such that $\chi(G-U_{0})=\chi(G-U_{1})=k-1$. Choose arbitrary elements $a_{1}\in A$ and $x_{1}\in L(a_{1})$. By Claim 2.1.3, for each $u\in U_{0}\cup U_{1}$, there is a unique element $y(u)\in L(u)$ adjacent to $x_{1}$ in $H$ (the uniqueness of $y(u)$ follows from the definition of a cover). Let

[TABLE]

Since $U_{0}$ and $U_{1}$ are independent sets in $G$, $I_{0}$ and $I_{1}$ are independent sets in $H$.

Claim 2.1.6**.**

There exists an element $a_{0}\in A\setminus\{a_{1}\}$ such that $L(a_{0})\cap N_{H}(I_{0})\not\subseteq N_{H}(x_{1})$.

Proof.

Assume that for all $a\in A\setminus\{a_{1}\}$, we have $L(a)\cap N_{H}(I_{0})\subseteq N_{H}(x_{1})$. Let $G^{\prime}\coloneqq G-U_{0}$, and for each $w\in V(G^{\prime})\cup A$, let $L^{\prime}(w)\coloneqq L(w)\setminus N_{H}(I_{0})$. By the definition of $I_{0}$, $L^{\prime}(a_{1})=L(a_{1})\setminus\{x_{1}\}$, so

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On the other hand, by our assumption, for each $a\in A\setminus\{a_{1}\}$, we have

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Since for all $v\in V(G)$, $\sigma(G^{\prime},A,L^{\prime},H,v)\leq\sigma(v)$, the minimality of $k$ implies the conclusion of Theorem 2.1 for $(k-1,G^{\prime},A,L^{\prime},H)$; in other words, the graph $\mathbf{J}(G^{\prime},A)$ is $(L^{\prime},H)$-colorable. By the definition of $L^{\prime}$, for any $(L^{\prime},H)$-coloring $I$ of $\mathbf{J}(G^{\prime},A)$, $I\cup I_{0}$ is an $(L,H)$-coloring of $\mathbf{J}(G,A)$; this is a contradiction. ∎

Using Claim 2.1.6, fix some $a_{0}\in A\setminus\{a_{1}\}$ satisfying $L(a_{0})\cap N_{H}(I_{0})\not\subseteq N_{H}(x_{1})$, and choose any

[TABLE]

Since $x_{0}\in N_{H}(I_{0})$, we can also choose $y_{0}\in I_{0}$ so that $x_{0}y_{0}\in E(H)$.

Claim 2.1.7**.**

$x_{0}\not\in N_{H}(I_{1})$.

Proof.

If there is $y_{1}\in I_{1}$ such that $x_{0}y_{1}\in E(H)$, then $x_{0}-y_{0}-x_{1}-y_{1}-x_{0}$ is a walk in $H$ whose existence is ruled out by Claim 2.1.5. ∎

Claim 2.1.8**.**

There is an element $a_{2}\in A\setminus\{a_{0},a_{1}\}$ such that $L(a_{2})\cap N_{H}(I_{1})\not\subseteq N_{H}(x_{0},x_{1})$.

Proof.

The proof is almost identical to the proof of Claim 2.1.6. Assume that for all $a\in A\setminus\{a_{0},a_{1}\}$, we have $L(a)\cap N_{H}(I_{1})\subseteq N_{H}(x_{0},x_{1})$. Let $G^{\prime}\coloneqq G-U_{1}$, $A^{\prime}\coloneqq A\setminus\{a_{0}\}$, and for each $w\in V(G^{\prime})\cup A^{\prime}$, let $L^{\prime}(w)\coloneqq L(w)\setminus N_{H}(\{x_{0}\}\cup I_{1})$. By the definition of $I_{1}$, $L(a_{1})\cap N_{H}(I_{1})=\{x_{1}\}$, so

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On the other hand, by our assumption, for each $a\in A\setminus\{a_{0},a_{1}\}$, we have

[TABLE]

Since for all $v\in V(G)$, $\sigma(G^{\prime},A^{\prime},L^{\prime},H,v)\leq\sigma(v)$, the minimality of $k$ implies the conclusion of Theorem 2.1 for $(k-1,G^{\prime},A^{\prime},L^{\prime},H)$; in other words, the graph $\mathbf{J}(G^{\prime},A^{\prime})$ is $(L^{\prime},H)$-colorable. By the definition of $L^{\prime}$, for any $(L^{\prime},H)$-coloring $I$ of $\mathbf{J}(G^{\prime},A)$, $I\cup\{x_{0}\}\cup I_{1}$ is an $(L,H)$-coloring of $\mathbf{J}(G,A)$. This is a contradiction. ∎

Now we are ready to finish the proof of Theorem 2.1. Fix some $a_{2}\in A\setminus\{a_{0},a_{1}\}$ satisfying $L(a_{2})\cap N_{H}(I_{1})\not\subseteq N_{H}(x_{0},x_{1})$, and choose any

[TABLE]

Since $x_{2}\in N_{H}(I_{1})$, there is $y_{1}\in I_{1}$ such that $x_{2}y_{1}\in E(H)$. Then $x_{0}-y_{0}-x_{1}-y_{1}-x_{2}$ is a walk in $H$ contradicting the conclusion of Claim 2.1.5. ∎

Now it is easy to derive Theorem 1.5. Indeed, let $G$ be a graph with $n$ vertices, $m$ edges, and chromatic number $k$, let $A$ be a finite set disjoint from $V(G)$, and let $(L,H)$ be a cover of $\mathbf{J}(G,A)$ such that for all $v\in V(G)$ and $a\in A$, $|L(v)|=|L(a)|=\chi(\mathbf{J}(G,A))=|A|+k$. Note that

[TABLE]

If $|A|\geq 3m$, then

[TABLE]

so Theorem 2.1 implies that $\mathbf{J}(G,A)$ is $(L,H)$-colorable, and hence $Z_{DP}(G)\leq 3m$. Moreover, if $\delta(G)\geq k-1$, then

[TABLE]

so $Z_{DP}(G)\leq 3m-\frac{3}{2}(k-1)n$, as desired. Finally, Corollary 1.6 follows from Theorem 1.5 and the fact that an $n$-vertex graph can have at most ${n\choose 2}\leq n^{2}/2$ edges.

3 Proof of Theorem 1.7

We will prove the following precise version of Theorem 1.7:

Theorem 3.1**.**

For all even $n\in\mathbb{N}$, $Z_{DP}(n)\geq n^{2}/4-n$.

Proof.

Let $n\in\mathbb{N}$ be even and let $k\coloneqq n/2-1$. Note that $n^{2}/4-n=k^{2}-1$. Thus, it is enough to exhibit an $n$-vertex bipartite graph $G$ and a cover $(L,H)$ of $\mathbf{J}(G,k^{2}-2)$ such that $|L(u)|=k^{2}$ for all $u\in V(\mathbf{J}(G,k^{2}-2))$, yet $\mathbf{J}(G,k^{2}-2)$ is not $(L,H)$-colorable.

Let $G\cong K_{n/2,n/2}$ be an $n$-vertex complete bipartite graph with parts $X=\{x,x_{0},\ldots,x_{k-1}\}$ and $Y=\{y,y_{0},\ldots,y_{k-1}\}$, where the indices [math], …, $k-1$ are viewed as elements of the additive group $\mathbb{Z}_{k}$ of integers modulo $k$. Let $A$ be a set of size $k^{2}-2$ disjoint from $X\cup Y$. For each $u\in X\cup Y\cup A$, let $L(u)\coloneqq\{u\}\times\mathbb{Z}_{k}\times\mathbb{Z}_{k}$. Let $H$ be the graph with vertex set $(X\cup Y\cup A)\times\mathbb{Z}_{k}\times\mathbb{Z}_{k}$ in which the following pairs of vertices are adjacent:

• –

$(u,i,j)$ and $(u,i^{\prime},j^{\prime})$ for all $u\in X\cup Y\cup A$ and $i$, $j$, $i^{\prime}$, $j^{\prime}\in\mathbb{Z}_{k}$ such that $(i,j)\neq(i^{\prime},j^{\prime})$;

• –

$(u,i,j)$ and $(v,i,j)$ for all $u\in\{x,y\}\cup A$, $v\in N_{\mathbf{J}(G,A)}(u)$, and $i$, $j\in\mathbb{Z}_{k}$;

• –

$(x_{s},i,j)$ and $(y_{t},i+s,j+t)$ for all $s$, $t$, $i$, $j\in\mathbb{Z}_{k}$.

It is easy to see that $(L,H)$ is a cover of $\mathbf{J}(G,A)$. We claim that $\mathbf{J}(G,A)$ is not $(L,H)$-colorable. Indeed, suppose that $I$ is an $(L,H)$-coloring of $\mathbf{J}(G,A)$. For each $u\in X\cup Y\cup A$, let $i(u)$ and $j(u)$ be the unique elements of $\mathbb{Z}_{k}$ such that $(u,i(u),j(u))\in I$. By the construction of $H$ and since $I$ is an independent set, we have

[TABLE]

for all $u\in X\cup Y$ and $a\in A$. Since all the $k^{2}-2$ pairs $(i(a),j(a))$ for $a\in A$ are pairwise distinct, $(i(u),j(u))$ can take at most $2$ distinct values as $u$ is ranging over $X\cup Y$. One of those $2$ values is $(i(y),j(y))$, and if $u\in X$, then

[TABLE]

so the value of $(i(u),j(u))$ must be the same for all $u\in X$; let us denote it by $(i,j)$. Similarly, the value of $(i(u),j(u))$ is the same for all $u\in Y$, and we denote it by $(i^{\prime},j^{\prime})$.

It remains to notice that the vertices $(x_{i^{\prime}-i},i,j)$ and $(y_{j^{\prime}-j},i^{\prime},j^{\prime})$ are adjacent in $H$, so $I$ is not an independent set. ∎

Now we can prove Corollary 1.8:

Proof of Corollary 1.8.

First, suppose that $G$ is an $n$-vertex graph with $\chi(G)=r$ that maximizes the difference $\chi_{DP}(G)-\chi(G)$. Adding edges to $G$ if necessary, we may arrange $G$ to be a complete $r$-partite graph. Assuming $2r>n$, at least $2r-n$ of the parts must be of size $1$, i.e., $G$ is of the form $\mathbf{J}(G^{\prime},2r-n)$ for some $2(n-r)$-vertex graph $G^{\prime}$. By Corollary 1.6, we have $\chi_{DP}(G)=\chi(G)$ as long as $2r-n\geq 6(n-r)^{2}$, which holds for all $r\geq n-(1/\sqrt{6}-o(1))\sqrt{n}$. This establishes the upper bound $r(n)\leq n-\Omega(\sqrt{n})$.

On the other hand, due to Theorem 1.7, for each $n$, we can find a graph $G$ with $s$ vertices, where $s\leq(2+o(1))\sqrt{n}$, such that $\chi_{DP}(\mathbf{J}(G,n-s))>\chi(\mathbf{J}(G,n-s))$. Since $\mathbf{J}(G,n-s)$ is an $n$-vertex graph, we get

[TABLE]

Acknowledgements.

The authors are grateful to the anonymous referees for their valuable comments and suggestions.