Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866918022174736384 |
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| author | Liu, Xiangge Jia, Caili Lu, Yong Zhong, Jiaxu |
| author_facet | Liu, Xiangge Jia, Caili Lu, Yong Zhong, Jiaxu |
| contents | The toughness of graph $G$, denoted by $τ(G)$, is $τ(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, 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. Recently, Benediktovich [Discrete Applied Mathematics. 365 (2025) 130--137] confirmed the Bondy's metaconjecture for $t$-tough graphs in the case when $t\in\{1;2;3\}$ in terms of the size, the spectral radius and the signless Laplacian spectral radius of the graph. In this paper, we will confirm the Bondy's metaconjecture for $t$-tough graphs in the case when $t\geq4$ in terms of the size, the spectral radius, the signless Laplacian spectral radius, the distance spectral radius and the distance signless Laplacian spectral radius of graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_11090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite Liu, Xiangge Jia, Caili Lu, Yong Zhong, Jiaxu Combinatorics The toughness of graph $G$, denoted by $τ(G)$, is $τ(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, 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. Recently, Benediktovich [Discrete Applied Mathematics. 365 (2025) 130--137] confirmed the Bondy's metaconjecture for $t$-tough graphs in the case when $t\in\{1;2;3\}$ in terms of the size, the spectral radius and the signless Laplacian spectral radius of the graph. In this paper, we will confirm the Bondy's metaconjecture for $t$-tough graphs in the case when $t\geq4$ in terms of the size, the spectral radius, the signless Laplacian spectral radius, the distance spectral radius and the distance signless Laplacian spectral radius of graphs. |
| title | Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.11090 |