Size Conditions for Pancyclicity of $t$-Tough Graphs

Abstract

The toughness of a connected noncomplete graph $G$ is
$$\tau(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subseteq V(G),\ c(G-S)\geq2\right\},$$
where $c(G)$ is the number of components of $G$; as usual, $\tau(K_n)=\infty$. In 1973, Bondy proposed the metaconjecture that almost every nontrivial condition implying Hamiltonicity should also imply pancyclicity, apart from a simple family of exceptional graphs. Recently, Benediktovich [Discrete Applied Mathematics 365 (2025), 130-137] confirmed Bondy's metaconjecture for $t$-tough graphs when $t\in\{1,2,3\}$ by using conditions on the size, the spectral radius, and the signless Laplacian spectral radius. This paper confirms Bondy's metaconjecture for $t$-tough graphs when $t\geq4$ by means of conditions on the size, the spectral radius, the signless Laplacian spectral radius, the distance spectral radius, and the distance signless Laplacian spectral radius. More precisely, if a $t$-tough graph $G$ has order $n>10t-3$ and size $m\geq\binom{n-2t}{2}+3t^2$, then $G$ is pancyclic.