Conflict-free coloring on closed neighborhoods of bounded degree graphs

03/12/2020

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by   Sriram Bhyravarapu, et al.

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The closed neighborhood conflict-free chromatic number of a graph G, denoted by χ_CN(G), is the minimum number of colors required to color the vertices of G such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos [Combin. Probab. Comput. 2009] showed that χ_CN(G) = O(log^2+εΔ), for any ε > 0, where Δ is the maximum degree. In [Combin. Probab. Comput. 2014], Glebov, Szabó and Tardos showed existence of graphs G with χ_CN(G) = Ω(log^2Δ). In this paper, we bridge the gap between the two bounds by showing that χ_CN(G) = O(log^2 Δ).