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arXiv · 2010.00063

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

Sriram Bhyravarapu·Subrahmanyam Kalyanasundaram
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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 $χ_{ON}(G)$. In previous work [WG 2020], we showed the upper bound $χ_{ON}(G) \leq dc(G) + 3$, where $dc(G)$ denotes the distance to cluster parameter of $G$. In this paper, we obtain the improved upper bound of $χ_{ON}(G) \leq dc(G) + 1$. We also exhibit a family of graphs for which $χ_{ON}(G) > dc(G)$, thereby demonstrating that our upper bound is tight.

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Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram. 2020-09-30. A Tight Bound for Conflict-free Coloring in terms of Distance to Cluster. https://arxiv.org/abs/2010.00063

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