A short note on conflict-free coloring on closed neighborhoods of bounded degree graphs

The closed neighborhood conflict-free chromatic number of a graph G, denoted by chi C N ( 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 showed that chi C N ( G ) = O ( log 2 + epsilon Delta ), for any epsilon > 0, where Delta is the maximum degree. In 2014, Glebov et al. showed existence of graphs G with chi C N ( G ) = omega ( log 2 Delta ). In this article, we bridge the gap between the two bounds by showing that chi C N ( G ) = O ( log 2 Delta ).