Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order $8$

TL;DR

This paper proves that the smallest tetravalent half-arc-transitive graphs with a dihedral vertex-stabiliser of order 8 have a certain minimal size, confirming no smaller examples exist.

Contribution

It establishes the minimal order for such graphs with dihedral vertex-stabiliser, answering a longstanding open question.

Findings

No smaller graphs with these properties exist.

The known example with 10752 vertices is minimal.

The result confirms the uniqueness of the known example.

Abstract

A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in [Conder and Maru\v{s}i\v{c}, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Combin. Theory Ser. B 88 (2003) 67--76]. This example has $10752$ vertices and vertex-stabiliser isomorphic to the dihedral group of order $8$. In this paper, we show that no such graphs of smaller order exist, thus answering a frequently asked question.