New Proofs of Results about Proper Conflict-Free Coloring of Graphs

A proper conflict-free k-coloring of a graph is a proper k-coloring in which each nonisolated vertex has a color that appears exactly once in its open neighborhood. A graph is PCF k-colorable if it admits a proper conflict-free k-coloring. The PCF chromatic number of a graph G, denoted by χpcf(G), is the minimum k such that G is PCF k-colorable. Caro et al conjectured that for a connected graph G with maximum degree Δ≥3, χpcfG≤Δ+1. One case in this conjecture, a connected graph with maximum degree 3 is PCF 4-colorable, can be derived from the result of Liu and Yu. Jiménez et al stated that the upper bound of PCF chromatic number of a graph G is max {5,χ(G)} without a proof. In this paper, we give new proofs of the two results above and derive that for a connected graph G with maximum degree Δ≥3, its complete subdivision is PCF (∆+1)-colorable.