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{$\epsilon$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that an $\epsilon$-fraction of the requests are satisfied. This notion was introduced in 2019 by Dvo\v{r}\'ak, 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 Lov\'asz 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 $\epsilon = \frac 15$ 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 Dvo\v{r}\'ak, Norin, and Postle regarding flexibility for list coloring in the setting of DP-coloring.