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

Feng, Yan-Quan ; Kwak, Jin Ho ; Zhou, Chuixiang

Journal of Algebraic Combinatorics, Tome 26 (2007) no. 4, pp. 431-451.

Résumé

Summary: 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-arc-transitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph $K _{4,4}$ 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.

Détail

Classification : 05C25, 20B25
Keywords: keywords graph, half-arc-transitive, tightly attached, covering