Single‐conflict colorings of degenerate graphs

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 ‐degenerate graph on vertices of edge‐multiplicity at most , colors are always enough to color the vertices of in a way that avoids every forbidden color pair. This answers a question of Dvořák, Esperet, Kang, and Ozeki for simple graphs.