Extremal Results on Conflict-free Coloring
Sriram Bhyravarapu, Shiwali Gupta, Subrahmanyam Kalyanasundaram, Rogers Mathew
TL;DR
This work investigates extremal aspects of conflict-free coloring in graphs, focusing on CFON and CFCN chromatic numbers $\chi_{ON}(G)$ and $\chi_{CN}(G)$. It extends prior line-graph results to broader claw-free classes, proving $\chi_{ON}(G) = O(k\log\Delta)$ and $\chi_{CN}(G) = O(\log k\log n)$ for $K_{1,k}$-free graphs, and develops a high-minimum-degree regime giving $\chi_{ON} = O(\ln^{1+\varepsilon}\Delta)$. A matching lower bound shows $f_{CN}(c\Delta^{1-\varepsilon}) = \Theta(\ln^2\Delta)$ for certain ranges of $\delta$, indicating near-tight behavior. The results combine probabilistic methods, hypergraph CF-coloring lemmas, and structural graph theory to map the landscape of extremal CF colorings and raise open questions about tighter bounds and broad applicability. These insights improve understanding of conflict-free coloring in geometric and sparse graph classes with potential applications in frequency assignment and related domains.
Abstract
A conflict-free open neighborhood coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph $G$, the smallest number of colors required for such a coloring is called the conflict-free open neighborhood (CFON) chromatic number and is denoted by $χ_{ON}(G)$. By considering closed neighborhood instead of open neighborhood, we obtain the analogous notions of conflict-free closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by $χ_{CN}(G)$). The notion of conflict-free coloring was introduced in 2002, and has since received considerable attention. In this paper, we study some extremal questions related to CFON and CFCN coloring.