Enumerative Chromatic Choosability

TL;DR

This paper studies enumerative chromatic-choosability, the property that the list color function $P_ ell(G,m)$ matches the chromatic polynomial $P(G,m)$ for all natural numbers $m$. Employing AM-GM-type bounds and DP-coloring techniques, the authors characterize the bipartite case ($hi(G)=2$) via the core of the graph, and show that the family $G=\Theta(2,2,2k)$ satisfies $P_ ell(G,m)=P(G,m)$ for all $m\ge3$, reinforcing the link between colorability and enumeration. They also demonstrate that $K_1\vee\Theta(2,2,2k)$ is enumeratively chromatic-choosable and provide a counterexample $K_{2,2,4}$ to illustrate that chromatic-choosability does not always imply weak enumerative chromatic-choosability. Together, these results illuminate when the enumerative and ordinary coloring counts coincide and motivate conjectures about joins and DP-coloring in broader graph classes.

Abstract

Chromatic-choosablility is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted $P_{\ell}(G,m)$, which is the list analogue of the chromatic polynomial of $G$, $P(G,m)$. It is known that for any graph $G$ there is a positive integer $k$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. In this paper, we study enumerative chromatic-choosability. A graph $G$ is enumeratively chromatic-choosable when $P_{\ell}(G,m) = P(G,m)$ whenever $m \in \mathbb{N}$. We completely determine the graphs of chromatic number two that are enumeratively chromatic-choosable. We construct examples of graphs that are chromatic-choosable but fail to be enumeratively-chromatic choosable, and finally, we explore a conjecture as to whether for every graph $G$, there is a $p \in \mathbb{N}$ such that the join of $G$ and $K_p$ is enumeratively chromatic-choosable. The techniques we use to prove results are diverse and include probabilistic ideas and ideas from DP (or correspondence)-coloring.

Figures (3)

• Figure 1: A 2-assignment for a copy of $\Theta(2,2,4)$ with exactly one proper coloring
• Figure 2: A 2-assignment for a copy of $\Theta(2,2,2k)$ with exactly one proper coloring
• Figure 3: The graph $G=K_{2,2,4}$ with its list assignment $L$.

Theorems & Definitions (43)

• Theorem 1: NR15
• Theorem 3: DZ22
• Theorem 4
• Theorem 5: KM18
• Conjecture 6
• Proposition 7
• Theorem 10
• Theorem 11
• Theorem 12
• Lemma 13