Remarks on proper conflict-free colorings of graphs

• Published in: Discrete mathematics Vol. 346; no. 2; p. 113221
• Main Authors: Caro, Yair, Petruševski, Mirko, Škrekovski, Riste
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2023
• Subjects: Conflict-free coloring; Neighborhood; PCF chromatic number; Planar graph; Proper coloring; Proper coloring; PCF chromatic number; Planar graph; Conflict-free coloring; Neighborhood
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2022.113221

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χpcf(G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χpcf(G) in terms of other graph parameters. In particular, we show that χpcf(G)≤5Δ(G)/2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4≤k≤6. The paper concludes with few open problems.