On the chromatic polynomial and counting DP-colorings of graphs

Kaul, Hemanshu; Mudrock, Jeffrey A.

doi:10.1016/j.aam.2020.102131

Cited by 20 publications

(71 citation statements)

References 34 publications

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“…However, for some graphs there are surprising differences. For example, similar to the list color function, P DP (G, m) = P (G, m) for every m ∈ N whenever G is chordal or an odd cycle [16]. On the other hand, we have the following two results.…”

Section: Theorem 1 ([30]mentioning

confidence: 89%

“…In 2015, Dvořák and Postle [11] introduced a generalization of list coloring called DPcoloring (they called it correspondence coloring) in order to prove that every planar graph without cycles of lengths 4 to 8 is 3-choosable. DP-coloring has been extensively studied over the past 5 years (see e.g., [3,4,5,6,7,8,16,17,18,19,22,23,25,26]). Intuitively, DP-coloring is a variation on list coloring where each vertex in the graph still gets a list of colors, but identification of which colors are different can change from edge to edge.…”

Section: List Coloring and Dp-coloringmentioning

confidence: 99%

“…In 2019, Kaul and the first author introduced a DP-coloring analogue of the chromatic polynomial in hopes of gaining a better understanding of DP-coloring and using it as a tool for making progress on some open questions related to the list color function [16]. Specifically, suppose H = (L, H) is a cover of graph G. Let P DP (G, H) be the number of H-colorings of G. Then, the DP color function of G, P DP (G, m), is the minimum value of P DP (G, H) where the minimum is taken over all possible m-fold covers H of G. 2 It is easy to show that for any graph G and m ∈ N, P DP (G, m) P (G, m) P (G, m).…”

Section: Theorem 1 ([30]mentioning

confidence: 99%

“…In studying the tightness of Theorem 3, the authors of [16] mentioned that if G is a unicyclic graph 4 on n vertices that contains a cycle of length 4, then P (G, m) − P DP (G, m) = Θ(m n−3 ). However, they stated that "we do not have an example of a graph G such that P (G, m) − P DP (G, m) = Θ(m n−2 )."…”

Section: Theorem 3 ([16]mentioning

confidence: 99%

“…However, they stated that "we do not have an example of a graph G such that P (G, m) − P DP (G, m) = Θ(m n−2 )." Motivated by a result of Bernshteyn, Kostochka, and Zhu [8] that says for any graph G there exists an N 3|E(G)| such that χ DP (K p ∨ G) = χ(K p ∨ G) whenever p N , the authors of [16] also studied P DP (K p ∨ G, m). Interestingly, it turns out that the question of whether there exist p, N ∈ N such that P DP (K p ∨ G, m) = P (K p ∨ G, m) whenever m N is related to the asymptotics of P (G, m) − P DP (G, m).…”

Section: Theorem 3 ([16]mentioning

confidence: 99%

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Answers to Two Questions on the DP Color Function

Mudrock

Seth²

2021

Electron. J. Combin.

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DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic polynomial of graph $G$ is denoted $P(G,m)$, and it is equal to the number of proper $m$-colorings of $G$. In 2019, Kaul and Mudrock introduced an analogue of the chromatic polynomial for DP-coloring; specifically, the DP color function of graph $G$ is denoted $P_{DP}(G,m)$. For vertex disjoint graphs $G$ and $H$, suppose $G \vee H$ denotes the join of $G$ and $H$. Two fundamental questions posed by Kaul and Mudrock are: (1) For any graph $G$ with $n$ vertices, is it the case that $P(G,m)-P_{DP}(G,m) = O(m^{n-3})$ as $m \rightarrow \infty$? and (2) For every graph $G$, does there exist $p,N \in \mathbb{N}$ such that $P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)$ whenever $m \geq N$? We show that the answer to both these questions is yes. In fact, we show the answer to (2) is yes even if we require $p=1$.

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Section: Theorem 1 ([30]mentioning

confidence: 89%