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Given an undirected graph $G = (V,E)$, a conflict-free coloring with respectto open neighborhoods (CFON coloring) is a vertex coloring such that everyvertex has a uniquely colored vertex in its open neighborhood. The minimumnumber 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 $\chi_{ON}(G) \leqdc(G) + 3$, where $dc(G)$ denotes the distance to cluster parameter of $G$. Inthis 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.

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