An algebraic approach for counting DP-3-colorings of sparse 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 the chromatic polynomial of a graph G, P(G,m), and the list color function, Pℓ(G,m), the DP-color function of G, denoted by PDP(G,m), counts the minimum number of DP-colorings over all possible m-fold covers. It follows that PDP(G,m)≤Pℓ(G,m)≤P(G,m). A function f is chromatic-adherent if for every graph G, f(G,a)=P(G,a) for some a≥χ(G) implies that f(G,m)=P(G,m) for all m≥a. It is known that the DP-color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose G is an n-vertex graph and H is a 3-fold cover of G. In this paper we associate with H a polynomial fG,H∈F3[x1,…,xn] so that the number of non-zeros of fG,H equals the number of H-colorings of G. We then use a well-known result of Alon and Füredi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on PDP(G,3) when 2n>|E(G)|. An easy consequence of this is that PDP(G,3)≥3n/6 for every n-vertex planar graph G of girth at least 5, improving the previously known bounds on both PDP(G,3) and Pℓ(G,3). Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP-color function.