Remarks on proper conflict-free colorings of graphs

TL;DR

This paper investigates the conflict-free vertex coloring of graphs, determining the minimum number of colors needed for various graph classes and establishing bounds and conditions for such colorings.

Contribution

It provides exact values for the PCF chromatic number of several basic graph classes and introduces new bounds and conditions for conflict-free colorings.

Findings

Exact PCF chromatic numbers for trees, cycles, hypercubes, and subdivisions.

Upper bounds on PCF chromatic number in terms of maximum degree.

Characterization of when the upper bounds are tight.

Abstract

A vertex coloring of a graph is said to be \textit{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., \textit{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 $\chi_{\mathrm{pcf}}(G)$. 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 $\chi_{\mathrm{pcf}}(G)$ in terms of other graph parameters. In particular, we show that $\chi_{\mathrm{pcf}}(G) \leq5\Delta(G)/2$ and characterize equality. Several sufficient conditions for PCF $k$-colorability of graphs are established for $4\le k\le 6$. The paper concludes with few open problems.