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arXiv · 2505.11090

Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite

Xiangge Liu·Caili Jia·Yong Lu·Jiaxu Zhong
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Abstract

The toughness of graph $G$, denoted by $\tau(G)$, is $\tau(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ 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.

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Xiangge Liu, Caili Jia, Yong Lu, Jiaxu Zhong. 2025-05-16. Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite. https://arxiv.org/abs/2505.11090

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