Half-Transitive Group Actions on Finite Graphs of Valency 4

The action of a subgroupGof automorphisms of a graphXis said to be 12-transitiveif it is vertex- and edge- but not arc-transitive. In this case the graphXis said to be (G, 12)-transitive. In particular,Xis 12-transitiveif it is (Aut X, 12)-transitive. The 12-transitive action ofGonXinduces an orientation of the edges ofXwhich is preserved byG. LetXhave valency 4. An even length cycleCinXis aG-alternating cycleif every other vertex ofCis the head and every other vertex ofCis the tail of its two incident edges in the above orientation. It transpires that allG-alternating cycles inXhave the same length and form a decomposition of the edge set ofX(Proposition 2.4); half of this length is denoted byrG(X) and is called theG-radiusofX. Moreover, it is shown that any two adjacentG-alternating cycles ofXintersect in the same number of vertices and that this number, called theG-attachment number aG(X) ofX, divides 2rG(X) (Proposition 2.6). IfXis 12-transitive, we let theradiusand theattachment numberofXbe, respectively, the Aut X-radius and the Aut X-attachment number ofX. The caseaG(X)=2rG(X) corresponds to the graphXconsisting of twoG-alternating cycles with the same vertex sets and leads to an arc-transitive circulant graph (Proposition 2.4). IfaG(X)=rG(X) we say that the graphXistightly G-attached. In particular, a 12-transitive graphXof valency 4 istightly attachedif it is tightly Aut X-attached. A complete classification of tightly attached 12-transitive graphs with odd radius and valency 4 is obtained (Theorem 3.4).