Half-Transitive Group Actions on Finite Graphs of Valency 4

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length cycle C in X is a G-alternating cycle if every other vertex of C is the head and every other vertex of C is the tail of its two incident edges in the above orientation. It transpires that all G-alternating cycles in X have the same length and form a decomposition of the edge set of X (Proposition 2.4); half of this length is denoted by rG(X ) and is called the G-radius of X. Moreover, it is shown that any two adjacent G-alternating cycles of X intersect in the same number of vertices and that this number, called the G-attachment number aG(X ) of X, divides 2rG(X ) (Proposition 2.6). If X is 1 2-transitive, we let the radius and the attachment number of X be, respectively, the Aut X-radius and the Aut X-attachment number of X. The case aG(X )=2rG(X ) corresponds to the graph X consisting of two G-alternating cycles with the same vertex sets and leads to an arc-transitive circulant graph (Proposition 2.4). If aG(X )=rG(X ) we say that the graph X is tightly G-attached. In particular, a 1 2-transitive graph X of valency 4 is tightly attached if it is tightly Aut X-attached. A complete classification of tightly attached 2 -transitive graphs with odd radius and valency 4 is obtained (Theorem 3.4). 1998 Academic Press