Spectral conditions of pancyclicity for t-tough graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910500113088512 |
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| author | Benediktovich, Vladimir I. |
| author_facet | Benediktovich, Vladimir I. |
| contents | More than 40 years ago Chvátal introduced a new graph invariant, which he called graph toughness. From then on a lot of research has been conducted, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. A pancyclic graph is certainly Hamiltonian, but not conversely. Bondy in 1976, however, suggested the "metaconjecture" that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. We confirm the Bondy conjecture for t-tough graphs in the case when $t\in \{ 1;2;3\}$ in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral conditions of pancyclicity for t-tough graphs Benediktovich, Vladimir I. Combinatorics Discrete Mathematics 05C45 (Primary), 05C50 (Secondary) More than 40 years ago Chvátal introduced a new graph invariant, which he called graph toughness. From then on a lot of research has been conducted, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. A pancyclic graph is certainly Hamiltonian, but not conversely. Bondy in 1976, however, suggested the "metaconjecture" that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. We confirm the Bondy conjecture for t-tough graphs in the case when $t\in \{ 1;2;3\}$ in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. |
| title | Spectral conditions of pancyclicity for t-tough graphs |
| topic | Combinatorics Discrete Mathematics 05C45 (Primary), 05C50 (Secondary) |
| url | https://arxiv.org/abs/2406.17089 |