A note on fractional DP-coloring of graphs

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that unlike the fractional list chromatic number, the fractional DP-chromatic number of a graph G, denoted χDP⁎(G), can be arbitrarily larger than χ⁎(G), the graph's fractional chromatic number. We generalize a result of Alon, Tuza, and Voigt (1997) on the fractional list chromatic number of odd cycles, and, in the process, show that for each k∈N, χDP⁎(C2k+1)=χ⁎(C2k+1). We also show that for any n≥2 and m∈N, if p⁎ is the solution in (0,1) to p=(1−p)n then χDP⁎(Kn,m)≤1/p⁎, and we prove a generalization of this result for multipartite graphs. Finally, we determine a lower bound on χDP⁎(K2,m) for any m≥3.