Tetravalent half-edge-transitive graphs and non-normal Cayley graphs

Xiuyun Wang (Beijing Jiaotong University, China)

A vertex-transitive graph X is called half-edge-transitive if its automorphism group Aut(X)  has two orbits of same  length on the arc-set  and two orbits on the edge-set of X.  We show that connected tetravalent half-edge-transitive graphs can have arbitrary large stabilizers.

We give a sufficient condition for non-normal Cayley graphs and by using the condition, infinitely many connected tetravalent non-normal Cayley graphs on non-abelian simple groups are constructed. As an application, all connected tetravalent non-normal Cayley graphs on the alternating group $A_6$ are determined.