Bounds for DP color function and canonical labelings

TL;DR

This paper establishes tight bounds for the DP color function of 2-connected graphs and explores the relationship between DP colorings and canonical labelings, revealing cases where the two concepts coincide or differ.

Contribution

It provides new upper bounds for the DP color function of 2-connected graphs and characterizes when DP colorings align with canonical labelings in specific graph classes.

Findings

Trees maximize the DP color function among connected graphs.

Tight upper bounds are derived for 2-connected graphs.

For unicyclic and theta graphs, DP colorings with the same polynomial have canonical labelings.

Abstract

The DP-coloring is a generalization of the list coloring, introduced by Dvo\v{r}\'{a}k and Postle. Let $\mathcal{H}=(L,H)$ be a cover of a graph $G$ and $P_{DP}(G,\mathcal{H})$ be the number of $\mathcal{H}$-colorings of $G$. The DP color function $P_{DP}(G,m)$ of $G$, introduced by Kaul and Mudrock, is the minimum value of $P_{DP}(G,\mathcal{H})$ where the minimum is taken over all possible $m$-fold covers $\mathcal{H}$ of $G$. For the family of $n$-vertex connected graphs, one can deduce that trees maximize the DP color function, from two results of Kaul and Mudrock. In this paper we obtain tight upper bounds for the DP color function of $n$-vertex $2$-connected graphs. Another concern in this paper is the canonical labeling in a cover. It is well known that if an $m$-fold cover $\mathcal{H}$ of a graph $G$ has a canonical labeling, then $P_{DP}(G,\mathcal{H})=P(G,m)$ in which $P(G,m)$ is the chromatic polynomial of $G$. However the converse statement of this conclusion is not always true. We give examples that for some $m$ and $G$, there exists an $m$-fold cover $\mathcal{H}$ of $G$ such that $P_{DP}(G,\mathcal{H})=P(G,m)$, but $\mathcal{H}$ has no canonical labelings. We also prove that when $G$ is a unicyclic graph or a theta graph, for each $m\geq 3$, if $P_{DP}(G,\mathcal{H})=P(G,m)$, then $\mathcal{H}$ has a canonical labeling.