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

TL;DR

This paper improves the upper bound for conflict-free coloring in graphs based on the distance to cluster parameter, showing it is at most one more than this parameter, and proves the bound is tight.

Contribution

The paper presents a tighter upper bound for conflict-free coloring in graphs in terms of the distance to cluster, improving previous results and establishing tightness.

Findings

Upper bound for CFON coloring is $oxed{ ext{dc}(G) + 1}$.

Constructed graphs show the bound is tight.

The bound improves upon previous $ ext{dc}(G) + 3$ result.

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.