Finite graphs of valency 4 and girth 4 admitting half-transitive group actions (original) (raw)

Finite graphs of valency 4 and girth 4 admitting 1/2-transitive group actions, that is, vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1/2-transitive if its automorphism group acts 1/2-transitively. There is a natural orientation of the edge set of a 1/2-transitive graph induced and preserved by its automorphism group. It is proved that in a finite 1/2-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to 2 .