On differences between DP-coloring and list coloring

TL;DR

This paper explores the differences between DP-coloring and list coloring, highlighting unique properties of DP-coloring through examples and bounds, and establishing new lower bounds for edge-DP-chromatic numbers.

Contribution

It identifies properties that distinguish DP-coloring from list coloring and provides new bounds for the edge-DP-chromatic number of regular graphs.

Findings

Existence of a planar bipartite graph with DP-chromatic number 4

Edge-DP-chromatic number of a d-regular graph is at least d+1 for d ≥ 2

DP-coloring exhibits properties not shared with list coloring

Abstract

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'{a}k and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.

Equations

$$V(H):=\{(u,c):u\in V(G)\text{ and }c\in L(u)\},$$

$$L^{\prime }(u):=\{(u,c):c\in L(u)\}.$$

$$I_f:=\{(u,f(u)):u\in V(G)\}$$

$$(u,f_I(u))\in I\cap L^{\prime }(u)$$

$$L^{\ast }(u):=\{\begin{array}{ll}{L}_{ij}(u)& \text{if }u\in V(Q_{ij});\\ \{x_1,x_2,x_3\}& \text{if }u=a^{\ast };\\ \{y_1,y_2,y_3\}& \text{if }u=b^{\ast }.\end{array}$$

$$\frac{dn}{2}-1=|E(G^{\prime })|=\sum _{c\in C}|M_c|⩽\frac{dn}{2}-2;$$

$$V(H):=E(G)\times {\mathbb{Z}}_d,$$

$${\chi }_{DP}(K_k^t)=\Delta (K_k^t)+1=tk-t+1.$$

$${\chi }_{DP}(\mathsf{MLine}(G))={\chi }_{DP}(K_d^2)=2d-1=2\Delta (G)-1.$$

Full text

1 Introduction

1.1 Basic notation and conventions

We use $\mathbb{N}$ to denote 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, except in Section 4, which mentions (loopless) multigraphs. For a graph $G$, $V(G)$ and $E(G)$ denote the vertex and the edge sets of $G$ respectively. For a subset $U\subseteq V(G)$, $G[U]$ is the subgraph of $G$ induced by $U$. For two subsets $U_{1}$, $U_{2}\subseteq V(G)$, $E_{G}(U_{1},U_{2})\subseteq E(G)$ is the set of all edges of $G$ with one endpoint in $U_{1}$ and the other one in $U_{2}$. The maximum degree of $G$ is denoted by $\Delta(G)$.

1.2 Graph coloring, list coloring, and DP-coloring

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

List coloring is a generalization of ordinary graph coloring that was introduced independently by Vizing [23] and Erdős, Rubin, and Taylor [13]. As in the case of ordinary graph coloring, let $C$ be a set of colors. A list assignment for a graph $G$ is a function $L\colon V(G)\to\operatorname{Pow}(C)$; if $|L(u)|=k$ for all $u\in V(G)$, then $L$ is called a $k$-list assignment. A proper coloring $f\colon V(G)\to C$ 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 $G$ admits an $L$-coloring for every $k$-list assignment $L$ for $G$. An immediate consequence of this definition is that $\chi_{\ell}(G)\geqslant\chi(G)$ for all graphs $G$, since ordinary coloring is the same as $L$-coloring with $L(u)=C$ for all $u\in V(G)$. On the other hand, it is well-known that the gap between $\chi(G)$ and $\chi_{\ell}(G)$ can be arbitrarily large; for instance, $\chi(K_{n,n})=2$, while $\chi_{\ell}(K_{n,n})=(1+o(1))\log_{2}(n)\to\infty$ as $n\to\infty$, where $K_{n,n}$ denotes the complete bipartite graph with both parts having size $n$.

In this paper we study a further generalization of list coloring that was recently introduced by Dvořák and Postle [12]; they called it correspondence coloring, and we call it DP-coloring for short. In the setting of DP-coloring, not only does each vertex get its own list of available colors, but also the identifications between the colors in the lists can vary from edge to edge.

Definition 1.1**.**

Let $G$ be a graph. A cover of $G$ is a pair $\mathscr{H}=(L,H)$, consisting of a graph $H$ and a function $L\colon V(G)\to\operatorname{Pow}(V(H))$, satisfying the following requirements:

1. (C1)

the sets $\{L(u)\,:\,u\in V(G)\}$ form a partition of $V(H)$; 2. (C2)

for every $u\in V(G)$, the graph $H[L(u)]$ is complete; 3. (C3)

if $E_{H}(L(u),L(v))\neq\varnothing$, then either $u=v$ or $uv\in E(G)$; 4. (C4)

if $uv\in E(G)$, then $E_{H}(L(u),L(v))$ is a matching.

A cover $\mathscr{H}=(L,H)$ of $G$ is $k$-fold if $|L(u)|=k$ for all $u\in V(G)$.

Remark 1.2**.**

The matching $E_{H}(L(u),L(v))$ in Definition 1.1(C4) does not have to be perfect and, in particular, is allowed to be empty.

Definition 1.3**.**

Let $G$ be a graph and let $\mathscr{H}=(L,H)$ be a cover of $G$. An $\mathscr{H}$-coloring of $G$ is an independent set in $H$ of size $|V(G)|$.

Remark 1.4**.**

By definition, if $\mathscr{H}=(L,H)$ is a cover of $G$, then $\{L(u)\,:\,u\in V(G)\}$ is a partition of $H$ into $|V(G)|$ cliques. Therefore, an independent set $I\subseteq V(H)$ is an $\mathscr{H}$-coloring of $G$ if and only if $|I\cap L(u)|=1$ for all $u\in V(G)$.

Definition 1.5**.**

Let $G$ be a graph. The DP-chromatic number $\chi_{DP}(G)$ of $G$ is the smallest $k\in\mathbb{N}$ such that $G$ admits an $\mathscr{H}$-coloring for every $k$-fold cover $\mathscr{H}$ of $G$.

Example 1.6**.**

Figure 1 shows two distinct $2$-fold covers of the $4$-cycle $C_{4}$. Note that $C_{4}$ admits an $\mathscr{H}_{1}$-coloring but not an $\mathscr{H}_{2}$-coloring. In particular, $\chi_{DP}(C_{4})\geqslant 3$; on the other hand, it can be easily seen that $\chi_{DP}(G)\leqslant\Delta(G)+1$ for any graph $G$, and so we have $\chi_{DP}(C_{4})=3$. A similar argument demonstrates that $\chi_{DP}(C_{n})=3$ for any cycle $C_{n}$ of length $n\geqslant 3$.

One can construct a cover of a graph $G$ based on a list assignment for $G$, thus showing that list coloring is a special case of DP-coloring and, in particular, $\chi_{DP}(G)\geqslant\chi_{\ell}(G)$ for all graphs $G$.

More precisely, let $G$ be a graph and suppose that $L\colon V(G)\to\operatorname{Pow}(C)$ is a list assignment for $G$, where $C$ is a set of colors. Let $H$ be the graph with vertex set

[TABLE]

in which two distinct vertices $(u,c)$ and $(v,d)$ are adjacent if and only if

• –

either $u=v$,

• –

or else, $uv\in E(G)$ and $c=d$.

For each $u\in V(G)$, set

[TABLE]

Then $\mathscr{H}\coloneqq(L^{\prime},H)$ is a cover of $G$, and there is a natural bijective correspondence between the $L$-colorings and the $\mathscr{H}$-colorings of $G$. Indeed, if $f\colon V(G)\to C$ is an $L$-coloring of $G$, then the set

[TABLE]

is an $\mathscr{H}$-coloring of $G$. Conversely, given an $\mathscr{H}$-coloring $I\subseteq V(H)$ of $G$, $|I\cap L^{\prime}(u)|=1$ for all $u\in V(G)$, so one can define an $L$-coloring $f_{I}\colon V(G)\to C$ by the property

[TABLE]

for all $u\in V(G)$.

1.3 DP-coloring vs. list coloring and the results of this note

Some upper bounds on list-chromatic number hold for DP-chromatic number as well. For instance, it is easy to see that $\chi_{DP}(G)\leqslant d+1$ for any $d$-degenerate graph $G$. Dvořák and Postle [12] observed that for any planar graph $G$, $\chi_{DP}(G)\leqslant 5$ and, moreover, $\chi_{DP}(G)\leqslant 3$ if $G$ is a planar graph of girth at least $5$ (these statements are extensions of classical results of Thomassen [20, 21] on list colorings).

Furthermore, there are statements about list coloring whose only known proofs involve DP/coloring in essential ways. For example, the reason why Dvořák and Postle originally introduced DP/coloring was to prove that every planar graph without cycles of lengths $4$ to $8$ is $3$-list-colorable [12, Theorem 1], thus answering a long-standing question of Borodin [7, Problem 8.1]. Another example can be found in [5], where Dirac’s theorem on the minimum number of edges in critical graphs [10, 11] is extended to the framework of DP-colorings, yielding a solution to the problem, posed by Kostochka and Stiebitz [17], of classifying list-critical graphs that satisfy Dirac’s bound with equality.

On the other hand, DP-coloring and list coloring are also strikingly different in some respects. For instance, Bernshteyn [3, Theorem 1.6] showed that the DP-chromatic number of every graph with average degree $d$ is $\Omega(d/\log d)$, i.e., close to linear in $d$. Recall that due to a celebrated result of Alon [2], the list-chromatic number of such graphs is $\Omega(\log d)$, and this bound is sharp for “small” bipartite graphs. In spite of this, known upper bounds on list-chromatic numbers often have the same order of magnitude as in the DP/coloring setting. For example, by Johansson’s theorem [16], triangle-free graphs $G$ of maximum degree $\Delta$ satisfy $\chi_{\ell}(G)=O(\Delta/\log\Delta)$. The same asymptotic upper bound holds for $\chi_{DP}(G)$ [3, Theorem 1.7]. Recently, Molloy [18] refined Johansson’s result to $\chi_{\ell}(G)\leqslant(1+o(1))\Delta/\ln\Delta$, and this improved bound, including the constant factor, also generalizes to DP-colorings [4].

Important tools in the study of list coloring that do not generalize to the framework of DP/coloring are the orientation theorems of Alon and Tarsi [1] and the closely related Bondy–Boppana–Siegel lemma (see [1]). Indeed, they can be used to prove that even cycles are $2$-list-colorable, while the DP-chromatic number of any cycle is $3$, regardless of its length (see Example 1.6). In this note we demonstrate the failure in the context of DP-coloring of two other list-coloring results whose proofs rely on either the Alon–Tarsi method or the Bondy–Boppana–Siegel lemma.

A well-known application of the orientation method is the following result:

Theorem 1.7** (Alon–Tarsi [1, Corollary 3.4]).**

Every planar bipartite graph is $3$-list-colorable.

We show that Theorem 1.7 does not hold for DP-colorings (note that every planar triangle-free graph is $3$-degenerate, hence $4$-DP-colorable):

Theorem 1.8**.**

There exists a planar bipartite graph $G$ with $\chi_{DP}(G)=4$.

This answers a question of Grytczuk (personal communication, 2016). We prove Theorem 1.8 in Section 2.

Our second result concerns edge colorings. Recall that the line graph $\mathsf{Line}(G)$ of a graph $G$ is the graph with vertex set $E(G)$ such that two vertices of $\mathsf{Line}(G)$ are adjacent if and only if the corresponding edges of $G$ share an endpoint. The chromatic number, the list-chromatic number, and the DP-chromatic number of $\mathsf{Line}(G)$ are called the chromatic index, the list-chromatic index, and the DP-chromatic index of $G$ and are denoted by $\chi^{\prime}(G)$, $\chi^{\prime}_{\ell}(G)$, and $\chi^{\prime}_{DP}(G)$ respectively. The following hypothesis is known as the Edge List Coloring Conjecture and is a major open problem in graph theory:

Conjecture 1.9** (Edge List Coloring Conjecture, see [15]).**

For every graph $G$, $\chi^{\prime}_{\ell}(G)=\chi^{\prime}(G)$.

In an elegant application of the orientation method, Galvin [14] verified the Edge List Coloring Conjecture for bipartite graphs:

Theorem 1.10** (Galvin [14]).**

For every bipartite graph $G$, $\chi^{\prime}_{\ell}(G)=\chi^{\prime}(G)=\Delta(G)$.

We show that this famous result fails for DP-coloring; in fact, it is impossible for a $d$-regular graph $G$ with $d\geqslant 2$ to have DP-chromatic index $d$:

Theorem 1.11**.**

If $d\geqslant 2$, then every $d$-regular graph $G$ satisfies $\chi^{\prime}_{DP}(G)\geqslant d+1$.

We prove Theorem 1.11 in Section 3.

Vizing [22] proved that the inequality $\chi^{\prime}(G)\leqslant\Delta(G)+1$ holds for all graphs $G$. He also conjectured the following weakening of the Edge List Coloring Conjecture:

Conjecture 1.12** (Vizing).**

For every graph $G$, $\chi^{\prime}_{\ell}(G)\leqslant\Delta(G)+1$.

We do not know if Conjecture 1.12 can be extended to DP-colorings:

Problem 1.13**.**

Do there exist graphs $G$ with $\chi^{\prime}_{DP}(G)\geqslant\Delta(G)+2$?

In Section 4 we discuss two natural ways to define edge-DP-colorings for multigraphs. According to one of them, the DP-chromatic index of the multigraph $K^{d}_{2}$ with two vertices joined by $d$ parallel edges is $2d$.

2 Proof of Theorem 1.8

In this section we construct a planar bipartite graph $G$ with DP-chromatic number $4$. The main building block of our construction is the graph $Q$ shown in Figure 3 on the left, i.e., the skeleton of the $3$-dimensional cube. Let $\mathscr{F}=(L,F)$ denote the cover of $Q$ shown in Figure 3 on the right.

Lemma 2.1**.**

The graph $Q$ is not $\mathscr{F}$-colorable.

Proof.

Suppose, towards a contradiction, that $I$ is an $\mathscr{F}$-coloring of $Q$. Since $L(a)=\{x\}$, we have $x\in I$, and, similarly, $y\in I$. Since $z_{1}$ is the only vertex in $L(c_{1})$ that is not adjacent to $x$ or $y$, we also have $z_{1}\in I$, and, similarly, $z_{2}\in I$. This leaves only $2$ vertices available in each of $L(d_{1})$, $L(d_{2})$, $L(d_{3})$, and $L(d_{4})$, and it is easy to see that these $8$ vertices do not contain an independent set of size $4$ (cf. the cover $\mathscr{H}_{2}$ of the $4$-cycle shown in Figure 1 on the right). ∎

Consider $9$ pairwise disjoint copies of $Q$, labeled $Q_{ij}$ for $1\leqslant i$, $j\leqslant 3$. For each vertex $u\in V(Q)$, its copy in $Q_{ij}$ is denoted by $u_{ij}$. Let $\mathscr{F}_{ij}=(L_{ij},F_{ij})$ be a cover of $Q_{ij}$ isomorphic to $\mathscr{F}$. Again, we assume that the graphs $F_{ij}$ are pairwise disjoint and use $u_{ij}$ to denote the copy of a vertex $u\in V(F)$ in $F_{ij}$. Let $G$ be the graph obtained from the (disjoint) union of the graphs $Q_{ij}$ by identifying the vertices $a_{11}$, …, $a_{33}$ to a new vertex $a^{\ast}$ and the vertices $b_{11}$, …, $b_{33}$ to a new vertex $b^{\ast}$. Let $H$ be the graph obtained from the union of the graphs $F_{ij}$ by identifying, for each $1\leqslant i$, $j\leqslant 3$, the vertices $x_{i1}$, $x_{i2}$, $x_{i3}$ to a new vertex $x_{i}$ and the vertices $y_{1j}$, $y_{2j}$, $y_{3j}$ to a new vertex $y_{j}$. Define the map $L^{\ast}\colon V(G)\to\operatorname{Pow}(V(H))$ as follows:

[TABLE]

Then $\mathscr{H}\coloneqq(L^{\ast},H)$ is a $3$-fold cover of $G$. We claim that $G$ is not $\mathscr{H}$-colorable. Indeed, suppose that $I$ is an $\mathscr{H}$-coloring of $G$ and let $i$ and $j$ be the indices such that $\{x_{i},y_{j}\}\subset I$. Then $I$ induces an $\mathscr{F}_{ij}$-coloring of $Q_{ij}$, which cannot exist by Lemma 2.1. Since $G$ is evidently planar and bipartite, the proof of Theorem 1.8 is complete.

3 Proof of Theorem 1.11

Let $d\geqslant 2$ and let $G$ be an $n$-vertex $d$-regular graph. If $\chi^{\prime}(G)=d+1$, then $\chi^{\prime}_{DP}(G)\geqslant d+1$ as well, so from now on we will assume that $\chi^{\prime}(G)=d$. In particular, $n$ is even. Indeed, a proper coloring of $\mathsf{Line}(G)$ is the same as a partition of $E(G)$ into matchings, and if $n$ is odd, then $d$ matchings can cover at most $d(n-1)/2<dn/2=|E(G)|$ edges of $G$.

Let $uv\in E(G)$ and let $G^{\prime}\coloneqq G-uv$. Our argument hinges on the following simple observation:

Lemma 3.1**.**

Let $C$ be a set of size $d$ and let $f\colon E(G^{\prime})\to C$ be a proper coloring of $\mathsf{Line}(G^{\prime})$. For each $w\in\{u,v\}$, let $f_{w}$ denote the unique color in $C$ not used in coloring the edges incident to $w$. Then $f_{u}=f_{v}$.

Proof.

For each $c\in C$, let $M_{c}\subseteq E(G^{\prime})$ denote the matching formed by the edges $e$ with $f(e)=c$. Then $|M_{c}|\leqslant n/2$ for all $c\in C$. Moreover, by definition, $\max\{|M_{f_{u}}|,|M_{f_{v}}|\}\leqslant n/2-1$. Thus, if $f_{u}\neq f_{v}$, then

[TABLE]

a contradiction. ∎

Let $\mathbb{Z}_{d}$ denote the additive group of integers modulo $d$ and let $H$ be the graph with vertex set

[TABLE]

in which the following pairs of vertices are adjacent:

• –

$(e,i)$ and $(e,j)$ for $e\in E(G)$ and $i$, $j\in\mathbb{Z}_{d}$ with $i\neq j$,

• –

$(e,i)$ and $(h,i)$ for $eh\in E(\mathsf{Line(G^{\prime})})$ and $i\in\mathbb{Z}_{d}$,

• –

$(uv,i)$ and $(uv^{\prime},i)$ for $uv^{\prime}\in E(G^{\prime})$ and $i\in\mathbb{Z}_{d}$;

• –

$(uv,i)$ and $(u^{\prime}v,i+1)$ for $u^{\prime}v\in E(G^{\prime})$ and $i\in\mathbb{Z}_{d}$.

For each $e\in E(G)$, let $L(e)\coloneqq\{e\}\times\mathbb{Z}_{d}$. Then $\mathscr{H}\coloneqq(L,H)$ is a $d$-fold cover of $\mathsf{Line}(G)$. We claim that $\mathsf{Line}(G)$ is not $\mathscr{H}$-colorable (which proves Theorem 1.11). Indeed, suppose that $I$ is an $\mathscr{H}$-coloring of $\mathsf{Line}(G)$. For each $e\in E(G^{\prime})$, let $f(e)$ denote the unique element of $\mathbb{Z}_{d}$ such that $(e,f(e))\in I$. Then $f$ is a proper coloring of $\mathsf{Line}(G^{\prime})$ with $\mathbb{Z}_{d}$ as its set of colors. Let $f_{u}$ be the unique element of $\mathbb{Z}_{d}$ that is not used in coloring the edges incident to $u$. Then the only element of $L(uv)$ that can, and therefore must, belong to $I$ is $(uv,i)$. On the other hand, Lemma 3.1 implies that $i$ is also the unique element of $\mathbb{Z}_{d}$ that is not used in coloring the edges incident to $v$, and, in particular, for some $u^{\prime}v\in E(G^{\prime})$, $f(u^{\prime}v)=i+1$. Since $(uv,i)$ and $(u^{\prime}v,i+1)$ are adjacent vertices of $H$, $I$ is not an independent set, which is a contradiction.

4 Edge-DP-colorings of multigraphs

One can extend the notion of DP-coloring to loopless multigraphs, see [6]. The definitions are almost identical; the only difference is that in Definition 1.1, (C4) is replaced by the following:

1. (C4*′*)

If $u$ and $v$ are connected by $t\geqslant 1$ edges in $G$, then $E_{H}(L(u),L(v))$ is a union of $t$ matchings.

An interesting property of DP-coloring of multigraphs is that the DP-chromatic number of a multigraph may be larger than its number of vertices. For example, the multigraph $K^{t}_{k}$ obtained from the complete graph $K_{k}$ by replacing each edge with $t$ parallel edges satisfies

[TABLE]

(See [6, Lemma 7].)

Similarly to the case of simple graphs, the line graph $\mathsf{Line}(G)$ of a multigraph $G$ is the graph with vertex set $E(G)$ such that two vertices of $\mathsf{Line}(G)$ are adjacent if and only if the corresponding edges of $G$ share at least one endpoint. Notice that, in particular, $\mathsf{Line}(G)$ is always a simple graph. Sometimes, instead of $\mathsf{Line}(G)$, it is more natural to consider the line multigraph $\mathsf{MLine}(G)$, where if two edges of $G$ share both endpoints, then the corresponding vertices of $\mathsf{MLine}(G)$ are joined by a pair edges. Line multigraphs were used, e.g., in the seminal paper by Galvin [14] and also in [8, 9].

Somewhat surprisingly, Shannon’s bound $\chi^{\prime}(G)\leqslant 3\Delta(G)/2$ [19] on the chromatic index of a multigraph $G$ does not extend to $\chi_{DP}(\mathsf{MLine}(G))$. Indeed, if $G\cong K^{d}_{2}$, i.e., if $G$ is the $2$-vertex multigraph with $d$ parallel edges, then $\mathsf{MLine}(G)\cong K^{2}_{d}$, so

[TABLE]

This is in contrast with the result in [8] that $\chi^{\prime}_{\ell}(G)\leqslant 3\Delta(G)/2$ for every multigraph $G$. However, we conjecture that the analog of Shannon’s theorem holds for line graphs:

Conjecture 4.1**.**

For every multigraph $G$, $\chi_{DP}(\mathsf{Line}(G))\leqslant 3\Delta(G)/2$.