Flexible $3$-valent graphs of even girth

TL;DR

This paper proves the existence of connected, flexible, 3-valent, vertex-transitive graphs with even girth for all integers, providing a constructive method for prime girth lengths.

Contribution

It establishes the existence of such graphs for all even girths and offers a constructive proof specifically for prime girth lengths.

Findings

Existence of connected flexible 3-valent vertex-transitive graphs for all even girths.

Constructive proof provided for prime girth lengths.

Applicable to all integers with even girth.

Abstract

We prove the existence of a connected flexible $3$-valent vertex-transitive graph of girth $2\ell$ for every integer $\ell$. We also give a constructive proof if $\ell$ is prime.