Some properties of minimally nonperfectly divisible graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911488107610112 |
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| author | Hu, Qiming Xu, Baogang Zhuang, Miaoxia |
| author_facet | Hu, Qiming Xu, Baogang Zhuang, Miaoxia |
| contents | A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$, and a graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. A clique $X$ of a connected graph $G$ is called a clique cutset if $G-X$ is disconnected. In this paper, we investigate the relationship between the perfect divisibility of a graph and its perfect weighted divisibility. We also show that $2P_3$-free or claw-free minimally nonperfectly divisible graphs contain no clique cutset, that conditionally answers a question of Hoàng [Discrete Math. \textbf{349} (2025) 114809]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01967 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some properties of minimally nonperfectly divisible graphs Hu, Qiming Xu, Baogang Zhuang, Miaoxia Combinatorics A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$, and a graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. A clique $X$ of a connected graph $G$ is called a clique cutset if $G-X$ is disconnected. In this paper, we investigate the relationship between the perfect divisibility of a graph and its perfect weighted divisibility. We also show that $2P_3$-free or claw-free minimally nonperfectly divisible graphs contain no clique cutset, that conditionally answers a question of Hoàng [Discrete Math. \textbf{349} (2025) 114809]. |
| title | Some properties of minimally nonperfectly divisible graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.01967 |