Constructing even radius tightly attached half-arc-transitive graphs of valency four

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41–76, 1998) classified odd radius tightly attached tetravalent half-arctransitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.