Chords of longest cycles in graphs with large circumferences

TL;DR

This paper proves longstanding conjectures that in certain large-circumference graphs, every longest cycle contains a chord, extending previous results and establishing new conditions for chords in longest cycles.

Contribution

The paper confirms Thomassen's and Harvey's conjectures for graphs with large circumferences and introduces a general result on chords in longest cycles containing a linear forest.

Findings

Confirmed Thomassen's conjecture for large-circumference graphs.

Confirmed Harvey's conjecture for large-circumference graphs.

Established a new result on chords in longest cycles with linear forests.

Abstract

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is a $2$-connected cubic graph, then any longest cycle must have a chord. He also showed that in any 3-connected graph with minimum degree at least four, some longest cycle must contain a chord. Harvey proved that every longest cycle has a chord for graphs with a large minimum degree. He also conjectured that any longest cycle in a 2-connected graph with minimum degree at least three has a chord. In this paper, we prove that both Thomassen's and Harvey's conjectures are true for graphs with large circumferences. We also prove a more general result for the existence of chords in longest cycles containing a linear forest.