The DP Color Function of Joins and Vertex-Gluings of Graphs

TL;DR

This paper investigates how the DP color function behaves under graph operations like joins and vertex-gluings, providing new insights and tools to understand its relationship with the chromatic polynomial.

Contribution

It introduces methods to analyze the DP color function for joins with complete graphs and vertex-gluings, advancing understanding of these operations' effects.

Findings

Determined the threshold where DP color function equals chromatic polynomial for certain graph constructions.

Developed new tools to study DP color function under graph joins and gluings.

Extended knowledge of DP color function beyond basic graph classes.

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial $P(G,m)$, the DP color function of a graph $G$, denoted $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. Chromatic polynomials for joins and vertex-gluings of graphs are well understood, but the effect of these graph operations on the DP color function is not known. In this paper we make progress on understanding the DP color function of the join of a graph with a complete graph and vertex-gluings of certain graphs. We also develop tools to study the DP color function under these graph operations, and we study the threshold (smallest $m$) beyond which the DP color function of a graph constructed with these operations equals its chromatic polynomial.