Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

Finite graphs of valency 4 and girth 4 admitting ½-transitive group actions, that is, vertex- and edge- but not arc-transitive group actions, are investigated. A graph is said to be ½-transitiveif its automorphism group acts ½-transitively. There is a natural orientation of the edge set of a ½-transitive graph induced and preserved by its automorphism group. It is proved that in a finite ½-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 Z2.