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

TL;DR

This paper extends the verification of Bondy's metaconjecture to $t$-tough graphs with $t extgreater=4$, establishing conditions under which such graphs are Hamiltonian, pancyclic, or bipartite based on various spectral parameters.

Contribution

It confirms Bondy's metaconjecture for $t$-tough graphs with $t extgreater=4$ using spectral graph theory measures, expanding previous results for smaller $t$ values.

Findings

Conditions on spectral radius imply Hamiltonicity and pancyclicity for $t$-tough graphs with $t extgreater=4$.

Spectral parameters can determine bipartiteness in $t$-tough graphs.

Extension of Bondy's conjecture verification to larger toughness values.

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.