A Note on Fractional DP-Coloring of Graphs

TL;DR

This paper explores fractional DP-coloring of graphs, showing specific cases where it equals fractional chromatic number and providing bounds for complete bipartite graphs and multipartite graphs, advancing understanding of this coloring variant.

Contribution

It generalizes a classical result on odd cycles and establishes bounds for fractional DP-chromatic numbers of bipartite and multipartite graphs.

Findings

Fractional DP-chromatic number equals fractional chromatic number for odd cycles.

Derived upper bounds for complete bipartite graphs.

Provided lower bounds for bipartite graphs with two vertices in one part.

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k 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 $\chi_{_{DP}}^*(G)$, can be arbitrarily larger than $\chi^*(G)$, 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 $k \in \mathbb{N}$, $\chi_{_{DP}}^*(C_{2k+1}) = \chi^*(C_{2k+1})$. We also show that for any $n \geq 2$ and $m \in \mathbb{N}$, if $p^*$ is the solution in $(0,1)$ to $p=(1-p)^n$ then $\chi_{_{DP}}^*(K_{n,m})\leq1/p^*$, and we prove a generalization of this result for multipartite graphs. Finally, we determine a lower bound on $\chi_{_{DP}}^*(K_{2,m})$ for any $m \geq 3$.