Tetravalent graphs admitting half-transitive group actions: Alternating cycles (Q1305529)

A graph \(X\) is said to be \((G,1/2)\)-transitive if the group \(G\) acts transitively on the vertices and edges, but not on the arcs of \(X\). The authors investigate finite, connected, 4-valent graphs \(X\) which are \((G,1/2)\)-transitive. It is proved that these graphs are covers of such graphs which have one of three attachment properties: tightly attached, loosely attached, or antipodally attached. Examples of these attachment properties are given and the complete classification of the tightly attached examples is presented.