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

The action of a subgroup GG of automorphisms of a graph XX is said to be half-arc-transitive   if it is vertex-transitive and edge-transitive but not arc-transitive. In this case the graph XX is said to be GG-half-arc-transitive  . In particular, when G=AutX the graph XX 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 ee is reachable   from an arc e′e′ if there exists an alternating path starting with ee and ending with e′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.