Single-conflict colorings of degenerate graphs

TL;DR

This paper studies a new graph coloring problem called single-conflict coloring, providing bounds on the number of colors needed for degenerate graphs with certain edge multiplicity, answering a previously open question.

Contribution

It establishes an upper bound on the number of colors needed for single-conflict coloring in degenerate graphs, solving an open problem for simple graphs.

Findings

O() colors suffice for certain degenerate graphs

Addresses an open question in graph coloring theory

Applicable to graphs with edge-multiplicity at most  log log n

Abstract

We consider the single-conflict coloring problem, a graph coloring problem in which each edge of a graph receives a forbidden ordered color pair. The task is to find a vertex coloring such that no two adjacent vertices receive a pair of colors forbidden at an edge joining them. We show that for any assignment of forbidden color pairs to the edges of a $d$-degenerate graph $G$ on $n$ vertices of edge-multiplicity at most $\log \log n$, $O(\sqrt{ d } \log n)$ colors are always enough to color the vertices of $G$ in a way that avoids every forbidden color pair. This answers a question of Dvo\v{r}\'ak, Esperet, Kang, and Ozeki for simple graphs (Journal of Graph Theory 2021).