Answers to Two Questions on the DP Color Function

TL;DR

This paper proves that the DP color function closely approximates the chromatic polynomial for large m and that for certain graph joins, the DP and classical chromatic polynomials eventually coincide, answering two open questions.

Contribution

It confirms that the difference between the chromatic polynomial and DP color function is bounded and that they agree beyond a certain point for specific graph constructions.

Findings

The difference P(G,m) - P_{DP}(G,m) is O(m^{n-3}) as m approaches infinity.

For any graph G, there exist p, N such that P_{DP}(K_p ∨ G, m) equals P(K_p ∨ G, m) for all m ≥ N.

Both fundamental questions posed by Kaul and Mudrock are affirmatively answered.

Abstract

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k 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 $P_{DP}(G,m)$. 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 $p=1$.