Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers

TL;DR

This paper constructs an infinite family of tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers of unbounded order, answering a long-standing open question in algebraic graph theory.

Contribution

It provides the first explicit construction of connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers of arbitrarily large order, expanding the known examples significantly.

Findings

Constructed graphs have vertex stabilizers of the form D8^2 × C2^m for all m ≥ 1.

The graphs exhibit numerous significant properties across different mathematical contexts.

Resolved the open problem of existence for such graphs with unbounded nonabelian stabilizers.

Abstract

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s\geqslant3$ has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer $\mathrm{D}_8^2\times\mathrm{C}_2^m$ for each integer $m\geqslant1$, where $\mathrm{D}_8^2$ is the direct product of two copies of the dihedral group of order $8$ and $\mathrm{C}_2^m$ is the direct product of $m$ copies of the cyclic group of order $2$. The graphs constructed have surprisingly many significant properties in various contexts.