Chords of longest cycles in graphs with large circumferences

TL;DR

This work addresses whether the longest cycles in graphs with large circumference necessarily contain chords, linking to Thomassen's and Harvey's longstanding conjectures. It proves that for a simple graph on $n$ vertices with minimum degree at least $3$ and circumference at least $c(G)\ge n-\frac{1+\sqrt{4n-3}}{2}$, every longest cycle has a chord, and it extends this chordality to cycles containing a linear forest or a specified edge (via Theorems main1–main3). The approach combines two elementary lemmas with a contraction-augmentation framework and sharp circumference bounds $f(n)$ and $g(n)$ to rule out chordless maximal cycles, yielding corollaries such as $c(G)\ge n-\sqrt{n}$. Overall, the paper advances the understanding of chordal structure in maximal cycles under large-circumference regimes and broadens the scope of Thomassen's and Harvey's conjectures beyond connectivity assumptions.

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.

Figures (1)

• Figure 1: A graph having a longest cycle with no chord and with circumference $5n\over 12$.

Theorems & Definitions (16)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Corollary 1.6
• Conjecture 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10