Perfect weighted divisibility is equivalent to perfect divisibility
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911392688242688 |
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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 $ω(H[B]) < ω(H)$. 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 $H$. In this paper, we prove that the perfect divisibility of a graph is equivalent to its perfect weighted divisibility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13695 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect weighted divisibility is equivalent to perfect divisibility 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 $ω(H[B]) < ω(H)$. 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 $H$. In this paper, we prove that the perfect divisibility of a graph is equivalent to its perfect weighted divisibility. |
| title | Perfect weighted divisibility is equivalent to perfect divisibility |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.13695 |