A Tight Bound for Conflict-free Coloring in terms of Distance to Cluster

Abstract

Given an undirected graph $G = (V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of $G$, denoted by $\chi_{ON}(G)$. In previous work [WG 2020], we showed the upper bound $\chi_{ON}(G) \leq dc(G) + 3$, where $dc(G)$ denotes the distance to cluster parameter of $G$. In this paper, we obtain the improved upper bound of $\chi_{ON}(G) \leq dc(G) + 1$. We also exhibit a family of graphs for which $\chi_{ON}(G) > dc(G)$, thereby demonstrating that our upper bound is tight.