Bounds and Hardness Results for Conflict-free Choosability

TL;DR

This paper investigates bounds and computational complexity of conflict-free coloring in hypergraphs, extending known results to list variants and analyzing the difficulty of related coloring problems.

Contribution

It extends an existing $O( ext{log}^2 ext{Delta})$ upper bound to partial list conflict-free coloring and explores complexity results for list conflict-free coloring variants.

Findings

Extended the $O( ext{log}^2 ext{Delta})$ bound to partial list conflict-free coloring.

Established complexity results for list open/closed-neighborhood conflict-free coloring.

Analyzed the relationship between conflict-free and partial conflict-free chromatic numbers.

Abstract

A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $\Delta$ is at most $O(\ln^2 \Delta)$. In this paper, we extend the $O(\ln^2 \Delta)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.