Odd coloring of $k$-trees
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909596850847744 |
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| author | Kashima, Masaki Ozeki, Kenta |
| author_facet | Kashima, Masaki Ozeki, Kenta |
| contents | An odd coloring of a graph is a proper coloring such that every non-isolated vertex has a color that appears at an odd number of its neighbors. This notion was introduced by Petrševski and Škrekovski in 2022. In this paper, we focus on odd coloring of $k$-trees, where a $k$-tree is a graph obtained from the complete graph of order $k+1$ by recursively adding a new vertex that is joined to a clique of order $k$ in the former graph. It follows from a result of Cranston, Lafferty, and Song in 2023 that every $k$-tree is odd $(2k+1)$-colorable. We improve this bound to show that every $k$-tree is odd $\left(k+2\left\lfloor\log_2 k\right\rfloor+3\right)$-colorable. Furthermore, when $k=2,3$, we show the tight bound that every 2-tree is odd $4$-colorable and that every 3-tree is odd $5$-colorable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Odd coloring of $k$-trees Kashima, Masaki Ozeki, Kenta Combinatorics 05C15 An odd coloring of a graph is a proper coloring such that every non-isolated vertex has a color that appears at an odd number of its neighbors. This notion was introduced by Petrševski and Škrekovski in 2022. In this paper, we focus on odd coloring of $k$-trees, where a $k$-tree is a graph obtained from the complete graph of order $k+1$ by recursively adding a new vertex that is joined to a clique of order $k$ in the former graph. It follows from a result of Cranston, Lafferty, and Song in 2023 that every $k$-tree is odd $(2k+1)$-colorable. We improve this bound to show that every $k$-tree is odd $\left(k+2\left\lfloor\log_2 k\right\rfloor+3\right)$-colorable. Furthermore, when $k=2,3$, we show the tight bound that every 2-tree is odd $4$-colorable and that every 3-tree is odd $5$-colorable. |
| title | Odd coloring of $k$-trees |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2504.20573 |