Fractional DP‐colorings of sparse graphs

Abstract

DP‐coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018), pp. 38–54]. In this paper we introduce and study the fractional DP‐chromatic number χDP*(G). We characterize all connected graphs G such that χDP*(G)⩽2: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt [Discrete Math. 165–166 (1997), pp. 31–38], the fractional list‐chromatic number χitalicℓ*(G) of any graph G equals its fractional chromatic number χ*(G). This equality does not extend to fractional DP‐colorings. Moreover, we show that the difference χDP*(G)−χ*(G) can be arbitrarily large, and, furthermore, χDP*(G)⩾d∕(2lnd) for every graph G of maximum average degree d⩾4. On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number.