J. Austral. Math. Soc.  73 (2002), 155-170
Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

Dragan Marusic
  IMFM, Oddelek za matematiko
  Univerza v Ljubljani
  Jadranska 19
  1000 Ljubljana
Roman Nedela
  Katedra Matematiky
  Univerzita Mateja Bela
  975 49 Banská Bystrica

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 $\mathbb{Z}_2$.
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