DP color functions versus chromatic polynomials

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.