Graphs with girth $2\ell+1$ and without longer odd holes that contain an odd $K_4$-subdivision
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909058872639488 |
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| author | Chen, Rong Zhou, Yidong |
| author_facet | Chen, Rong Zhou, Yidong |
| contents | We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is 3-colorable. Recently, Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and $\mathcal{G}_3$ is 3-colorable. In this paper, we prove that no $4$-vertex-critical graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ has an odd $K_4$-subdivision. Using this result, Chen proved that all graphs in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ are 3-colorable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12376 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Graphs with girth $2\ell+1$ and without longer odd holes that contain an odd $K_4$-subdivision Chen, Rong Zhou, Yidong Combinatorics 05C15, 05C17, 05C69 We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is 3-colorable. Recently, Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and $\mathcal{G}_3$ is 3-colorable. In this paper, we prove that no $4$-vertex-critical graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ has an odd $K_4$-subdivision. Using this result, Chen proved that all graphs in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ are 3-colorable. |
| title | Graphs with girth $2\ell+1$ and without longer odd holes that contain an odd $K_4$-subdivision |
| topic | Combinatorics 05C15, 05C17, 05C69 |
| url | https://arxiv.org/abs/2210.12376 |