Constructing infinitely many half-arc-transitive covers of tetravalent graphs

TL;DR

This paper proves the existence of infinitely many specific graph covers and classifies vertex stabilizers in finite tetravalent half-arc-transitive graphs, advancing understanding of their symmetry properties.

Contribution

It introduces a method to construct infinitely many half-arc-transitive covers and classifies vertex stabilizers up to order 256 in these graphs.

Findings

Existence of infinitely many half-arc-transitive covers for certain graphs

Classification of vertex stabilizers up to order 256

New insights into the symmetry structure of tetravalent graphs

Abstract

We prove that, given a finite graph $\Sigma$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $\Sigma$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.