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arXiv · 2410.02336

On strong odd colorings of graphs

Yair Caro·Mirko Petruševski·Riste Škrekovski·Zsolt Tuza
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Abstract

A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is colored by $c$. The smallest integer $k$ for which $G$ admits a strong odd coloring with $k$ colors is the strong odd chromatic number, $\chi_{soc}(G)$. These coloring notion and graph parameter were recently defined in [H. Kwon and B. Park, Strong odd coloring of sparse graphs, ArXiv:2401.11653v2]. We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs and graph products.

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Yair Caro, Mirko Petruševski, Riste Škrekovski, Zsolt Tuza. 2024-10-03. On strong odd colorings of graphs. https://arxiv.org/abs/2410.02336

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