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

Bhyravarapu, Sriram and Kalyanasundaram, Subrahmanyam (2022) A tight bound for conflict-free coloring in terms of distance to cluster. Discrete Mathematics, 345 (11). ISSN 0012-365X

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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 χON(G). In previous work [WG 2020], we showed the upper bound χON(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 χON(G)≤dc(G)+1. We also exhibit a family of graphs for which χON(G)>dc(G), thereby demonstrating that our upper bound is tight. © 2022 Elsevier B.V.

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