Asymptotically Optimal Proper Conflict‐Free Coloring

ABSTRACT A proper conflict‐free coloring of a graph is a coloring of the vertices such that any two adjacent vertices receive different colors, and for every non‐isolated vertex , some color appears exactly once on the neighborhood of . Caro, Petruševski and Škrekovski conjectured that every connected graph with maximum degree has a proper conflict‐free coloring with at most colors. This conjecture holds for and remains open for . In this article we prove that this conjecture holds asymptotically; namely, every graph with maximum degree has a proper conflict‐free coloring with colors.