Single-conflict colorings of degenerate graphs
Peter Bradshaw, Tomáš Masařík
TL;DR
This work studies single-conflict colorings, where each edge forbids a specific ordered color pair, and introduces a probabilistic framework on oriented graphs with conflict functions. It distinguishes uniquely restrictive conflicts and general conflicts (via a notion of restrictiveness) and applies the Lovász Local Lemma to obtain color bounds. The main results give a degeneracy-aware bound $\chi_{\nleftrightarrow}(G) \le \lceil \sqrt{d}\,2^{\mu/2+2}\sqrt{\mu}\sqrt{1+\log((d+1)\Delta)} \rceil$ for $d$-degenerate graphs of maximum degree $\Delta$ and edge multiplicity $\mu$, implying the simple-graph bound $\chi_{\nleftrightarrow}(G) \le \lceil 2\sqrt{d\,[1+\log((d+1)\Delta)]} \rceil$ and a cooperative-coloring result for families of degenerate graphs. The results resolve the question of Dvořák et al. for simple graphs and illuminate the role of edge multiplicity and degeneracy in single-conflict colorings, while also providing a framework that explains known DP-coloring connections and yields generalizations. Limitations include logarithmic factors and a negative example showing the bound cannot be improved beyond certain growth in edge multiplicity.
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 Dvořák, Esperet, Kang, and Ozeki for simple graphs (Journal of Graph Theory 2021).