DP color functions versus chromatic polynomials for hypergraphs (I)

TL;DR

This paper explores the differences between DP color functions and chromatic polynomials in hypergraphs, focusing on how cycle length and joins with complete graphs affect their relationship, revealing conditions where they differ or coincide.

Contribution

It establishes new conditions under which the DP color function is strictly less than the chromatic polynomial and when they are equal for hypergraph joins with complete graphs.

Findings

For hypergraphs with even girth, DP color functions are eventually less than chromatic polynomials.

Certain hypergraphs with specific edge and cycle properties have identical DP and chromatic functions beyond a threshold.

Equality of DP and chromatic functions holds for uniform hypergraph joins with complete graphs beyond some size.

Abstract

For a hypergraph $\mathcal{H}$, the DP color function $P_{DP}(\mathcal{H},k)$ of $\mathcal{H}$ is an extension of the chromatic polynomial $P(\mathcal{H},k)$ with the property that $P_{DP}(\mathcal{H},k) \le P(\mathcal{H},k)$ for all positive integers $k$. In this article, we primarily investigate the influence of the minimum cycle length on the DP-coloring function, as well as the relevant properties of the DP-coloring function of $\mathcal{H} \vee K_p$ (i.e., the join of $\mathcal{H}$ and $K_p$). We show that for any linear and uniform hypergraph $\mathcal H$ with even girth, there exists a positive integer $N$ such that $P_{DP} (\mathcal H, k) < P(\mathcal H, k)$ for all integers $k\ge N$, and this conclusion also holds for any hypergraph $\mathcal{H}$ that contains an edge $e$ with the properties that $\mathcal{H}-e$ has exactly $|e|-1$ components and any shortest cycle in $\mathcal{H}$ containing $e$ is an even cycle. For the hypergraph $\mathcal{H}\vee K_p$, we prove that if $\mathcal{H}$ is uniform, then there exist positive integers $p$ and $N$ such that $P_{DP}(\mathcal{H} \vee K_p,k)=P(\mathcal{H} \vee K_p,k)$ holds for all integers $k\geq N$.