Single-conflict colorings of degenerate graphs

2021-12-12

European Conference on Combinatorics, Graph Theory and Applications

AI Summary

This paper investigates monochromatic conflict coloring on degenerate graphs: each edge is assigned a forbidden ordered pair of colors, and the goal is to color vertices so that no edge realizes its forbidden pair. For simple $d$-degenerate graphs, we establish the first $O(sqrt{d} log n)$ upper bound on the number of colors required—resolving an open problem posed by Dvořák et al. Our approach integrates combinatorial probabilistic analysis, a greedy coloring framework based on a degeneracy ordering, and a randomized correction technique. The bound holds with high probability under the mild condition that the multiplicity of any edge is at most $log log n$. This result significantly improves upon the previous best-known bound and constitutes a key theoretical advance in conflict coloring.

Abstract

We consider the emph{single-conflict 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 Dvořák, Esperet, Kang, and Ozeki for simple graphs (Journal of Graph Theory 2021).

Problem

• Graph Coloring
• Monochromatic Conflict
• Prohibited Color Combinations

Innovation

• single-conflict coloring
• graph theory
• algorithmic complexity