Seymour and Woodall's conjecture holds for graphs with independence number two
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913697429979136 |
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| author | Chen, Rong Deng, Zijian |
| author_facet | Chen, Rong Deng, Zijian |
| contents | Woodall (and Seymour independently) in 2001 proposed a conjecture that every graph $G$ contains every complete bipartite graph on $χ(G)$ vertices as a minor, where $χ(G)$ is the chromatic number of $G$. In this paper, we prove that for each positive integer $\ell$ with $2\ell \leq χ(G)$, each graph $G$ with independence number two contains a $K^{\ell}_{\ell,χ(G)-\ell}$-minor, implying that Seymour and Woodall's conjecture holds for graphs with independence number two, where $K^{\ell}_{\ell,χ(G)-\ell}$ is the graph obtained from $K_{\ell,χ(G)-\ell}$ by making every pair of vertices on the side of the bipartition of size $\ell$ adjacent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Seymour and Woodall's conjecture holds for graphs with independence number two Chen, Rong Deng, Zijian Combinatorics Woodall (and Seymour independently) in 2001 proposed a conjecture that every graph $G$ contains every complete bipartite graph on $χ(G)$ vertices as a minor, where $χ(G)$ is the chromatic number of $G$. In this paper, we prove that for each positive integer $\ell$ with $2\ell \leq χ(G)$, each graph $G$ with independence number two contains a $K^{\ell}_{\ell,χ(G)-\ell}$-minor, implying that Seymour and Woodall's conjecture holds for graphs with independence number two, where $K^{\ell}_{\ell,χ(G)-\ell}$ is the graph obtained from $K_{\ell,χ(G)-\ell}$ by making every pair of vertices on the side of the bipartition of size $\ell$ adjacent. |
| title | Seymour and Woodall's conjecture holds for graphs with independence number two |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.02643 |