Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909539121496064 |
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| author | Liu, Feng |
| author_facet | Liu, Feng |
| contents | Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been independently confirmed by Xu, Li, and Zhou, as well as by Ota and Sanka. Inspired by this, we prove that every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected Liu, Feng Combinatorics 05C42, 05C45 Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been independently confirmed by Xu, Li, and Zhou, as well as by Ota and Sanka. Inspired by this, we prove that every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected. |
| title | Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected |
| topic | Combinatorics 05C42, 05C45 |
| url | https://arxiv.org/abs/2503.12860 |