Hamilton and long cycles in $t$-tough graphs with $t>1$

TL;DR

This paper proves that for t-tough graphs with t>1, either a large cycle exists or the graph is the Petersen graph, extending understanding of cycle lengths in tough graphs.

Contribution

It establishes a new cycle length bound for t-tough graphs with t>1, characterizing when the Petersen graph is the only exception.

Findings

If G is a t-tough graph with t>1, then G has a cycle of length at least min{n, 2δ+4} or G is the Petersen graph.

The result generalizes previous bounds on cycle lengths in tough graphs.

The Petersen graph is uniquely characterized as the exception in this cycle length condition.

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 $\min\{n,2\delta+4\}$ or $G$ is the Petersen graph.