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

Abstract

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 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.

Tetravalent half-arc-transitive graphs of order 8p

“…B Xiuyun Wang wangxiuyun732@126.com In view of the fact that 4 is the smallest admissible valency for a half-arc-transitive graph, special attention has rightly been given to the study of tetravalent half-arctransitive graphs. In particular, constructing and classifying the tetravalent half-arctransitive graphs is currently one of active topics in algebraic graph theory (for example, see [1,4,[10][11][12][13]19,20,24]). Let p be a prime.…”

Tetravalent half-arc-transitive graphs of order pqr

“…In view of the fact that 4 is the smallest admissible valency for a halfarc-transitive graph, special attention has rightly been given to the study of tetravalent half-arc-transitive graphs. In particular, constructing and classifying the tetravalent half-arc-transitive graphs is currently one of active topics in algebraic graph theory (for example, see [7][8][9] and [10][11][12][13]). …”

Characterization of edge‐transitive 4‐valent bicirculants

“…Trivalent edge‐transitive bicirculants have been classified in a series of articles (see , , , ). As for the 4‐valent ones, several partial cases were considered in , , , , , , , , , . In this article, we settle the missing cases and thus give a complete characterization of 4‐valent edge‐transitive bicirculants.…”