Half-transitive graphs of valency 4 with prescribed attachment numbers

A graph X is said to be 1/2-transitive if its automorphism group Aut X acts vertex- and edge-, but not are-transitively on X. Then Aut X induces an or ientation of the edges of X. If X has valency 4, then this orientation give s rise to so-called alternating cycles, that is even length cycles in X who se every other vertex is the head and every other vertex is the tail of its two incident edges in the above orientation. All alternating cycles have t he same length 2r(X), where r(X) is the radius of X, and any two adjacent a lternating cycles intersect in the same number of vertices, called the atta chment number a(X) of X. All known examples of 1/2-transitive graphs have a ttachment number 1, r or 2r, where r is the radius of the graph. In this ar ticle, we construct 1/2-transitive graphs with all other possible attachmen t numbers. The case of attachment number 2 is dealt with in more detail. (C ) 2000 John Wiley & Sons, Inc.