An infinite family of half-arc-transitive graphs with universal reachability relation

The action of a subgroup G of automorphisms of a graph X is said to be half-arc-transitive if it is vertex-transitive and edge-transitive but not arc-transitive. In this case the graph X is said to be G -half-arc-transitive. In particular, when G = Aut X the graph X is said to be half-arc-transitive. Two oppositely oriented digraphs may be associated with any such graph in a natural way. The reachability relation of a graph admitting a half-arc-transitive group action is an equivalence relation defined on either of these two digraphs as follows. An arc e is reachable from an arc e ′ if there exists an alternating path starting with e and ending with e ′ . The reachability relation is clearly universal for all vertex-primitive half-arc-transitive graphs. The smallest known vertex-primitive half-arc-transitive graphs have valency 14 and no such graphs of valency smaller than 10 exist. The natural framework for the question of the existence of half-arc-transitive graphs with universal reachability relation is therefore the family of vertex-imprimitive half-arc-transitive graphs, and in particular those of valency less than 14. It is known that no such graph of valency 4 exists (see D. Marušič, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41–76). In this paper an infinite family of vertex-imprimitive half-arc-transitive graphs of valency 12 with universal reachability relation is constructed. These graphs have a solvable automorphism group but are not metacirculants.