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arXiv · 2203.01088

Remarks on proper conflict-free colorings of graphs

Yair Caro·Mirko Petruševski·Riste Škrekovski
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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.

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Yair Caro, Mirko Petruševski, Riste Škrekovski. 2022-03-02. Remarks on proper conflict-free colorings of graphs. https://arxiv.org/abs/2203.01088

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