Fractional DP-Colorings of Sparse Graphs

TL;DR

This paper introduces the fractional DP-chromatic number for graphs, characterizes certain graphs with low fractional DP-chromatic number, and explores the differences and bounds compared to fractional list-chromatic number.

Contribution

It defines and studies the fractional DP-chromatic number, characterizes graphs with fractional DP-chromatic number at most 2, and analyzes bounds and differences from fractional list-chromatic number.

Findings

Graphs with no odd cycles and at most one even cycle have fractional DP-chromatic number ≤ 2

The difference between fractional DP-chromatic and list-chromatic numbers can be arbitrarily large

For graphs with maximum average degree d ≥ 4, fractional DP-chromatic number ≥ d/(2 ln d)

Abstract

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $\chi_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $\chi_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $\chi^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $\chi^\ast_{DP}(G) - \chi^\ast(G)$ can be arbitrarily large, and, furthermore, $\chi^\ast_{DP}(G) \geq d/(2 \ln d)$ for every graph $G$ of maximum average degree $d \geq 4$. 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.