Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908865009811456 |
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| author | Ozeki, Kenta Zhang, Leilei |
| author_facet | Ozeki, Kenta Zhang, Leilei |
| contents | Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vrána, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs Ozeki, Kenta Zhang, Leilei Combinatorics Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vrána, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian. |
| title | Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.03361 |