Odd 4-coloring of outerplanar graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910568830468096 |
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| author | Kashima, Masaki Zhu, Xuding |
| author_facet | Kashima, Masaki Zhu, Xuding |
| contents | 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 Petruševski and Škrekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petruševski, and Škrekovski 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, Petruševski, and Škrekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Odd 4-coloring of outerplanar graphs Kashima, Masaki Zhu, Xuding Combinatorics 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 Petruševski and Škrekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petruševski, and Škrekovski 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, Petruševski, and Škrekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs. |
| title | Odd 4-coloring of outerplanar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.19362 |