Half-Arc-Transitive Graphs and the Fano Plane

A subgroup G of the automorphism group of a graph Gamma acts half-arc-transitively on Gamma if the natural actions of G on the vertex-set and edge-set of Gamma are both transitive, but the natural action of G on the arc-set of Gamma is not transitive. When G = Aut(Gamma) the graph Gamma is said to be half-arc-transitive. Given a bipartite cubic graph with a certain degree of symmetry two covering constructions that provide infinitely many tetravalent graphs admitting half-arc-transitive groups of automorphisms are introduced. Symmetry properties of constructed graphs are investigated. In the second part of the paper the two constructions are applied to the Heawood graph, the well-known incidence graph of the Fano plane. It is proved that the members of the infinite family resulting from one of the two constructions are all half-arc-transitive, and that the infinite family resulting from the second construction contains a mysterious family of arc-transitive graphs that emerged within the classification of tightly attached half-arc-transitive graphs of valence 4 back in 1998 and 2008.