Tetravalent graphs admitting half-transitive group actions: Alternating cycles

In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not tran sitive on the arcs of X. Such a graph X is said to be (G, 1/2)-transitive. The group G induces an orientation of the edges of X, and a certain class o f cycles of X (called alternating cycles) determined by the group G is iden tified as having an important influence on the structure of X. The alternat ing cycles are those in which consecutive edges have opposite orientations. It is shown that X is a cover of a finite, connected, r-valent, (G, 1/2)-t ransitive graph for which the alternating cycles have one of three addition al special properties, namely they are tightly attached, loosely attached, or antipodally attached. We give examples with each of these special attach ment properties and moreover we complete the classification (begun in a sep arate paper by the first author) of the tightly attached examples. (C) 1999 Academic Press.