A Deletion-Contraction Relation for the DP Color Function

TL;DR

This paper introduces a deletion-contraction relation for the DP color function, extending its definition to multigraphs and defining a dual DP color function to establish new bounds on DP colorings.

Contribution

It presents a novel deletion-contraction relation for the DP color function and introduces the dual DP color function, expanding the theoretical framework for DP-coloring.

Findings

Established a deletion-contraction relation for the DP color function.

Extended the DP color function to multigraphs.

Derived a new lower bound on the DP color function using the dual DP color function.

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 $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool for computing the chromatic polynomial of graph $G$ is the deletion-contraction formula which relates $P(G,m)$ to the chromatic polynomials of two smaller graphs. The DP color function of a graph $G$, denoted $P_{DP}(G,m)$, is a DP-coloring analogue of the chromatic polynomial, and $P_{DP}(G,m)$ is the minimum number of DP-colorings of $G$ over all possible $m$-fold covers. In this paper we present a deletion-contraction relation for the DP color function. To make this possible, we extend the definition of the DP color function to multigraphs. We also introduce the dual DP color function of a graph $G$, denoted $P^*_{DP}(G,m)$, which counts the maximum number of DP-colorings of $G$ over certain $m$-fold covers. We show how the dual DP color function along with our deletion-contraction relation yields a new general lower bound on the DP color function of a graph.