The DP Color Function of Bipartite Graphs

Abstract

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of $P(G,q)$, the chromatic polynomial of a graph $G$, the DP color function of $G$, denoted by $P_{DP}(G,q)$, counts the minimum number of DP-colorings over all $q$-fold covers of $G$. It follows that $P_{DP}(G,q)\le P(G,q)$. It is known that there are graphs for which $P_{DP}(G,q)<P(G,q)$ for all sufficiently large $q$; in fact, all bipartite graphs containing a cycle have this property. A fundamental open question about DP color functions asks whether, for every graph $G$, there exist $N\in \mathbb{N}$ and a polynomial $p$ such that $P_{DP}(G,q)=p(q)$ whenever $q\ge N$. In this paper we answer this question affirmatively for all bipartite graphs. Specifically, if $G$ is an $n$-vertex bipartite graph with $c$ components, then $P_{DP}(G,q)=(-1{)}^{n-c}{q}^c{T}_G(1-q,1)$ for all sufficiently large $q$, where $T_G(x,y)$ is the Tutte polynomial of $G$. The ideas we develop also yield an asymptotic formula for $P(G,q)-P_{DP}(G,q)$ whenever the girth of $G$ is even.

17 pages, 1 figure