Bounds and Hardness Results for Conflict-free Choosability
Shiwali Gupta, Rogers Mathew
TL;DR
The paper advances the theory of conflict-free coloring by extending partial/list frameworks to open and closed neighborhood variants, deriving powerful upper bounds and establishing hardness results. It generalizes Pach–Tardos type probabilistic methods to list CF coloring, yielding $ch^*_{CF}(\mathcal{H}) = O(t \Gamma^{1/t} \ln \Gamma)$ and, for graphs with maximum degree $\Delta$, $ch^*_{ON}(G)$ and $ch^*_{CN}(G)$ both bound by $O(\ln^2 \Delta)$. It further shows that several conflict-free choosability problems are $\Pi_2^P$-complete on bipartite, planar triangle-free, and planar graphs via constructions $H_G, H'_G$, and $H^k_G$, while noting polynomial-time solvability for 1-CFON-choosability. Together, these results deepen understanding of list-based CF coloring and delineate complexity boundaries for practical network-choosability tasks. The work also identifies open questions on small-k cases and tightness gaps between variants, guiding future research in combinatorial coloring and complexity.
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 $Δ$ is at most $O(\ln^2 Δ)$. In this paper, we extend the $O(\ln^2 Δ)$ 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.