The perfect divisibility and chromatic number of some odd hole-free graphs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918381366542336 |
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| author | He, Weihua Shi, Yueping Wu, Rong Yao, Zheng-an |
| author_facet | He, Weihua Shi, Yueping Wu, Rong Yao, Zheng-an |
| contents | A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least one edge admits a partition of $V(H)$ into sets $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. $G$ is short-holed if every hole in $G$ has length 4. A hammer is the graph obtained by identifying one vertex of a $K_3$ and one end vertex of a $P_3$.
In this paper, we prove that (i) (odd hole, hammer, $K_{2,3}$)-free graphs are perfectly divisible, (ii) $χ(G)\le 4ω(G)(ω(G)-1)$ if $G$ is short-holed and $(K_1+C_4)$-free, (iii) $χ(G)\le 2ω(G)-1$ if $G$ is short-holed and $(K_1\cup K_3)$-free, and (iv) $χ(G)\le 16ω(G)-24$ if $G$ is short-holed and $(K_1+(K_1\cup K_3))$-free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The perfect divisibility and chromatic number of some odd hole-free graphs He, Weihua Shi, Yueping Wu, Rong Yao, Zheng-an Combinatorics A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least one edge admits a partition of $V(H)$ into sets $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. $G$ is short-holed if every hole in $G$ has length 4. A hammer is the graph obtained by identifying one vertex of a $K_3$ and one end vertex of a $P_3$. In this paper, we prove that (i) (odd hole, hammer, $K_{2,3}$)-free graphs are perfectly divisible, (ii) $χ(G)\le 4ω(G)(ω(G)-1)$ if $G$ is short-holed and $(K_1+C_4)$-free, (iii) $χ(G)\le 2ω(G)-1$ if $G$ is short-holed and $(K_1\cup K_3)$-free, and (iv) $χ(G)\le 16ω(G)-24$ if $G$ is short-holed and $(K_1+(K_1\cup K_3))$-free. |
| title | The perfect divisibility and chromatic number of some odd hole-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.09549 |