DP-Coloring Cartesian Products of Graphs

TL;DR

This paper investigates the DP-chromatic number of Cartesian products of graphs, establishing bounds, tools for lower bounds, and exploring the role of the DP color function in understanding these products.

Contribution

It introduces bounds for the DP-chromatic number of Cartesian products and develops tools for lower bound arguments, advancing the understanding of DP-coloring in graph products.

Findings

Established an upper bound for $ ext{chi}_{DP}(G oxempty H)$.

Demonstrated the sharpness of the bounds and their variants.

Highlighted the importance of the DP color function in product graph coloring.

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Motivated by results related to list coloring Cartesian products of graphs, we initiate the study of the DP-chromatic number, $\chi_{DP}$, of the same. We show that $\chi_{DP}(G \square H) \leq \text{min}\{\chi_{DP}(G) + \text{col}(H), \chi_{DP}(H) + \text{col}(G) \} - 1$ where $\text{col}(H)$ is the coloring number of the graph $H$. We focus on building tools for lower bound arguments for $\chi_{DP}(G \square H)$ and use them to show the sharpness of the bound above and its various forms. Our results illustrate that the DP color function of $G$, the DP analogue of the chromatic polynomial, is essential in the study of the DP-chromatic number of the Cartesian product of graphs, including the following question that extends the sharpness problem above and the classical result on gap between list chromatic number and chromatic number: given any graph $G$ and $k \in \mathbb{N}$, what is the smallest $t$ for which $\chi_{DP}(G \square K_{k,t})= \chi_{DP}(G) + k$?