A tight bound for conflict-free coloring in terms of distance to cluster

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) <= 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) < 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. (C) 2022 Elsevier B.V. All rights reserved.