Longest cycles and longest chordless cycles in $2$-connected graphs

TL;DR

This paper explores conjectures relating the lengths of longest cycles and chordless cycles in 2-connected graphs, proving the conjecture for specific classes and establishing equivalences involving wheel graphs.

Contribution

It introduces a new conjecture involving wheels, proves Harvey's conjecture for certain graph classes, and links these conjectures to properties of wheel graphs and $ ext{ell}$-holed graphs.

Findings

Harvey's conjecture holds for $ ext{ell}$-holed graphs.

Harvey's conjecture holds for graphs with small induced circumference.

Thomassen's conjecture is confirmed for these classes.

Abstract

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of its longest chordless cycles, respectively. In $2017$, Harvey proposed a stronger conjecture: Every $2$-connected graph $G$ with minimum degree at least $3$ has $c(G)\geq c'(G)+2$. This conjecture implies Thomassen's chord conjecture. We observe that wheels are the unique hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus we need only consider non-hamiltonian graphs for Harvey's conjecture. In this paper, we propose a conjecture involving wheels that is equivalent to Harvey's conjecture on non-hamiltonian graphs. A graph is $\ell$-holed if its all holes have length exactly $\ell$. Furthermore, we prove that Harvey's conjecture holds for $\ell$-holed graphs and graphs with a small induced circumference. Consequently, Thomassen's conjecture also holds for this two classes of graphs.