Flexible DP 3-coloring of sparse multigraphs

Abstract

A \emph{request} on a graph assigns a preferred color to a subset of the vertices. A graph $G$ is \emph{$\hat{I}\mu$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that an $\hat{I}\mu$-fraction of the requests are satisfied. This notion was introduced in 2019 by Dvořák, Norin, and Postle, who also proved important properties of flexible colorings and posed several natural problems. However, the weighted version of this problem is a special case of the much older problem of fractional hypergraph matchings, introduced by Lovász in 1975. We study flexibly DP-colorable multigraphs. We prove that every loopless multigraph with maximum average degree less than $3$ is $\frac{1}{5}$-flexibly DP $3$-colorable, except for an infinite family of multigraphs that we completely characterize. The constant $\hat{I}\mu =\frac{1}{5}$ is best possible in the weighted setting, as shown by an infinite family of tight examples. Our result follows from a stronger statement in terms of potential. We also provide a family of graphs that gives a negative answer to a question by Dvořák, Norin, and Postle regarding flexibility for list coloring in the setting of DP-coloring.

39 pages