Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

In this paper we study finite, connected, 4-valent graphsXwhich admit an action of a groupGwhich is transitive on vertices and edges, but not transitive on the arcs ofX. Such a graphXis said to be (G, 1/2)-transitive. The groupGinduces an orientation of the edges ofX, and a certain class of cycles ofX(called alternating cycles) determined by the groupGis identified as having an important influence on the structure ofX. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown thatXis a cover of a finite, connected, 4-valent, (G, 1/2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they aretightly attached, loosely attached, orantipodally attached.We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples.