DP color functions versus chromatic polynomials (II)

TL;DR

This paper investigates the relationship between DP color functions and chromatic polynomials, providing conditions under which graphs belong to sets where these functions are equal or differ, with applications to specific graph classes.

Contribution

It introduces new criteria based on cycle lengths and spanning trees to classify graphs in the sets $DP_{\approx}$ and $DP_{<}$, extending previous results.

Findings

Graphs with certain cycle length conditions belong to $DP_{\approx}$.

Graphs with specific even cycle length conditions belong to $DP_{<}$.

Applications include plane near-triangulations and complete multipartite graphs.

Abstract

For any connected graph $G$, let $P(G,m)$ and $P_{DP}(G,m)$ denote the chromatic polynomial and DP color function of $G$, respectively. It is known that $P_{DP}(G,m)\le P(G,m)$ holds for every positive integer $m$. Let $DP_\approx$ (resp. $DP_<$) be the set of graphs $G$ for which there exists an integer $M$ such that $P_{DP}(G,m)=P(G,m)$ (resp. $P_{DP}(G,m)<P(G,m)$) holds for all integers $m \ge M$. Determining the sets $DP_\approx$ and $DP_<$ is a key problem on the study of the DP color function. For any edge set $E_0$ of $G$, let $\ell_G(E_0)$ be the length of a shortest cycle $C$ in $G$ such that $|E(C)\cap E_0|$ is odd whenever such a cycle exists, and $\ell_G(E_0)=\infty$ otherwise. Write $\ell_G(E_0)$ as $\ell_G(e)$ if $E_0=\{e\}$. In this paper, we prove that if $G$ has a spanning tree $T$ such that $\ell_G(e)$ is odd for each $e\in E(G)\setminus E(T)$, the edges in $E(G)\setminus E(T)$ can be labeled as $e_1,e_2,\cdots, e_q$ with $\ell_G(e_i)\le \ell_G(e_{i+1})$ for all $1\le i\le q-1$ and each edge $e_i$ is contained in a cycle $C_i$ of length $\ell_G(e_i)$ with $E(C_i)\subseteq E(T)\cup \{e_j: 1\le j\le i\}$, then $G$ is a graph in $DP_{\approx}$. As a direct application of this conclusion, all plane near-triangulations and complete multipartite graphs with at least three partite sets belong to $DP_{\approx}$. We also show that if $E^*$ is an edge set of $G$ such that $\ell_{G}(E^*)$ is even and $E^*$ satisfies certain conditions, then $G$ belongs to $DP_<$. In particular, if $\ell_G(E^*)=4$, where $E^*$ is a set of edges between two disjoint vertex subsets of $G$, then $G$ belongs to $DP_<$. Both results extend known ones in [DP color functions versus chromatic polynomials, $Advances\ in\ Applied\ Mathematics$ 134 (2022), article 102301].