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

TL;DR

The paper extends Bondy\'s metaconjecture to $t$-tough graphs with $t\ge4$ by establishing both size- and spectral-condition sufficiencies for Hamiltonicity and, in fact, pancyclic or bipartite structure. It shows that a simple connected $t$-tough graph on $n>10t-3$ vertices with $m\ge\binom{n-2t}{2}+3t^2$ is Hamiltonian, and thus pancyclic or bipartite, and then translates spectral bounds on $\lambda_1(G)$, $q(G)$, $\lambda_1(D(G))$, and $\eta_1(G)$ into the same edge-count threshold to derive corresponding sufficient conditions. The key contributions are Theorem 1.1 (a size-based criterion) and Theorem 1.2 (spectral-based criteria), which together broaden the applicability of Bondy\'s metaconjecture to higher toughness. This work strengthens the link between graph toughness, Hamiltonian properties, and spectral graph theory, with potential implications for network design and cycle-structure analysis in robust graphs.

Abstract

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.

Figures (1)

• Figure 1: Corresponding graphs in the proof of $t=4$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Corollary 2.6
• Lemma 2.7
• Lemma 2.8