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arXiv · 2407.19362

Odd 4-coloring of outerplanar graphs

Masaki Kashima·Xuding Zhu
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Abstract

A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petru\v{s}evski and \v{S}krekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petru\v{s}evski, and \v{S}krekovski showed that every outerplanar graph is odd 5-colorable, and this bound is tight since the cycle of length 5 is not odd 4-colorable. Recently, the first author and others showed that every maximal outerplanar graph is odd 4-colorable. In this paper, we show that a connected outerplanar graph $G$ is odd 4-colorable if and only if $G$ contains a block which is not a copy of the cycle of length 5. This strengthens the result by Caro, Petru\v{s}evski, and \v{S}krekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs.

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Masaki Kashima, Xuding Zhu. 2024-07-28. Odd 4-coloring of outerplanar graphs. https://arxiv.org/abs/2407.19362

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