Fractional Strict Degeneracy of Graphs

TL;DR

This paper introduces two new degeneracy analogues for fractional DP-coloring, providing upper bounds for the fractional DP-chromatic number of various graph classes.

Contribution

It extends degeneracy concepts to fractional DP-coloring, offering bounds for fractional DP-chromatic numbers of specific graph families.

Findings

Bound the fractional DP-chromatic number of unicyclic graphs.

Bound the fractional DP-chromatic number of some complete bipartite graphs.

Bound the fractional DP-chromatic number of sparse graphs.

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. The DP-chromatic number of a graph $G$, $\chi_{_{DP}}(G)$, is the analogue of the chromatic number of $G$ in the DP context and is bounded above by the degeneracy of $G$ plus one. Over the last two years a plethora of authors have introduced variations on the notion of degeneracy and used these new ideas to give improved bounds on the DP-chromatic number of certain families of graphs. Fractional DP-coloring is a generalization of fractional list coloring introduced by Bernshteyn, Kostochka, and Zhu in 2019. In this paper we introduce two analogues of the degeneracy of a graph to the fractional context, each of which bound its fractional DP-chromatic number from above. We use these analogues to bound the fractional DP-chromatic number of a variety of graphs including unicyclic graphs, some complete bipartite graphs, and sparse graphs.