On the girth and connectivity of cubic graphs with a unique longest cycle

TL;DR

This paper constructs an infinite family of 2-connected cubic graphs with girth 5 that have a unique longest cycle, challenging assumptions about cycle structure in such graphs.

Contribution

It introduces a new infinite family of cubic, 2-connected, non-Hamiltonian graphs with girth 5, featuring a unique longest cycle.

Findings

Existence of infinite family of such graphs.

These graphs are non-Hamiltonian with girth 5.

They contain a unique longest cycle.

Abstract

We show that there exists an infinite family of cubic $2$-connected non-hamiltonian graphs with girth $5$ containing a unique longest cycle.