On tetravalent half-arc-transitive graphs of girth 5

TL;DR

This paper investigates tetravalent half-arc-transitive graphs of girth 5, characterizing their structure, properties, and providing classifications and infinite families, extending previous work on graphs of smaller girth.

Contribution

It extends the classification of tetravalent half-arc-transitive graphs to girth 5, analyzing their cycles, properties, and providing new infinite families and classifications.

Findings

Most such graphs have directed 5-cycles

The 5-cycles are consistent cycles for the automorphism group

Provided classifications and infinite families of these graphs

Abstract

A subgroup of the automorphism group of a graph $\G$ is said to be {\em half-arc-transitive} on $\G$ if its action on $\G$ is transitive on the vertex set of $\G$ and on the edge set of $\G$ but not on the arc set of $\G$. Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth $5$ admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth $5$.