The Praeger-Xu graphs: Cycle structures, maps and semitransitive orientations

Abstract

We consider tetravalent graphs within a family introduced by Praeger and Xu in 1989. These graphs have the property of having exceptionally large symmetry groups among all tetravalent graphs. This very property makes them unsuitable for the use of simple computer techniques. We apply techniques from coding theory to determine for which values of the parameters the graphs allow cycle structures, semitransitive orientations, or rotary maps; all without recourse to the use of computers.