DP color functions versus chromatic polynomials

TL;DR

This paper investigates the relationship between DP color functions and chromatic polynomials, establishing conditions under which they are equal or differ, and extends previous results with new graph-theoretic criteria.

Contribution

It generalizes prior findings by identifying new graph conditions that determine when DP color functions match or are less than chromatic polynomials.

Findings

If an edge's shortest cycle length is even, then P_{DP}(G)<P(G).

Graphs with a certain spanning tree structure satisfy P_{DP}(G)≈P(G).

Counterexamples show the converse does not always hold.

Abstract

For any graph $G$, the chromatic polynomial of $G$ is the function $P(G,m)$ which counts the number of proper $m$-colorings of $G$ for each positive integer $m$. The DP color function $P_{DP}(G,m)$ of $G$, introduced by Kaul and Mudrock in 2019, is a generalization of $P(G,m)$ with $P_{DP}(G,m)\le P(G,m)$ for each positive integer $m$. Let $P_{DP}(G)\approx P(G)$ (resp. $P_{DP}(G)< P(G)$) denote the property that $P_{DP}(G,m)=P(G,m)$ (resp. $P_{DP}(G,m)<P(G,m)$) holds for sufficiently large integers $m$.It is an interesting problem of finding graphs $G$ for which $P_{DP}(G)\approx P(G)$ (resp. $P_{DP}(G,m)<P(G,m)$) holds. Kaul and Mudrock showed that if $G$ has an even girth, then $P_{DP}(G)<P(G)$ and Mudrock and Thomason recently proved that $P_{DP}(G)\approx P(G)$ holds for each graph $G$ which has a dominating vertex. We shall generalize their results in this article. For each edge $e$ in $G$, let $\ell(e)=\infty$ if $e$ is a bridge of $G$, and let $\ell(e)$ be the length of a shortest cycle in $G$ containing $e$ otherwise. We first show that if $\ell(e)$ is even for some edge $e$ in $G$, then $P_{DP}(G)<P(G)$ holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that $P_{DP}(G)\approx P(G)$ holds for every graph $G$ that contains a spanning tree $T$ such that for each $e\in E(G)\setminus E(T)$, $\ell(e)$ is odd and $e$ contained in a cycle $C$ of length $\ell (e)$ with the property that $\ell(e')<\ell(e)$ for each $e'\in E(C)\setminus (E(T)\cup \{e\})$. Some open problems are proposed in this article.