A Sharp Bound for the Circumference in $t$-tough graphs with $t>1$

TL;DR

This paper establishes a sharp lower bound on the circumference of t-tough graphs with t>1, showing that such graphs either contain a long cycle or are the Petersen graph, thus advancing understanding of graph toughness and cycle length.

Contribution

It provides a precise bound on the circumference of t-tough graphs for t>1, characterizing the extremal case as the Petersen graph.

Findings

Graphs with t>1 have a cycle length at least min{n, 2δ+5} or are the Petersen graph.

The Petersen graph is the unique extremal case for the bound.

The result sharpens previous bounds on cycle length in t-tough graphs.

Abstract

It is proved that if $G$ is a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$ then either $G$ has a cycle of length at least $\min\{n,2\delta+5\}$ or $G$ is the Petersen graph.