Paper

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

Xia, Binzhou

2020

Overview

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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Graph theory / Graphs / Group theory / Questions / Riemann surfaces