A classification of tetravalent edge-transitive metacirculants of odd order

TL;DR

This paper classifies tetravalent edge-transitive metacirculants of odd order, showing most are normal graphs except for four known exceptions, and identifies conditions under which they are half-transitive.

Contribution

It provides a complete classification of tetravalent edge-transitive metacirculants of odd order, including their normality and half-transitivity properties.

Findings

Most tetravalent edge-transitive metacirculants are normal graphs.

Four known graphs are exceptions to normality.

Cayley graphs of non-abelian metacyclic groups are half-transitive.

Abstract

In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant $\Gamma$ is a normal graph except for four known graphs. If further, $\Gamma$ is a Cayley graph of a non-abelian metacyclic group, then $\Gamma$ is half-transitive.