Combinatorial Nullstellensatz and DP-coloring of graphs

Abstract

We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon–Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari et al. (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph G to satisfy χDP(G)≤3, and we completely determine the DP-chromatic number of squares of all cycles.