Long cycles in Hamiltonian graphs

TL;DR

This paper proves that Hamiltonian graphs with minimum degree at least 3 contain nearly spanning cycles, confirming an old conjecture asymptotically, using a mix of constructive and non-constructive methods.

Contribution

It establishes an asymptotic version of Sheehan's 1975 conjecture on cycle lengths in Hamiltonian graphs with minimum degree three.

Findings

Hamiltonian graphs with minimum degree ≥ 3 contain cycles of length n-o(n)

Confirms Sheehan's conjecture asymptotically

Introduces combined constructive and parity-based techniques

Abstract

We prove that if an $n$-vertex graph with minimum degree at least $3$ contains a Hamiltonian cycle, then it contains another cycle of length $n-o(n)$; this implies, in particular, that a well-known conjecture of Sheehan from 1975 holds asymptotically. Our methods, which combine constructive, poset-based techniques and non-constructive, parity-based arguments, may be of independent interest.