A �-transitive graph of valency 4 with a nonsolvable group of automorphisms

Journal of Combinatorial Theory, Series B

Kwak

2007

103

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K 3,3 and the s-regular cyclic or elementary abelian coverings of the complete graph K 4 for each s 1 are classified when the fibrepreserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4p 2 or 6p 2 is given for each s 1 and each prime p.

Section: Introductionmentioning

confidence: 99%

Cubic symmetric graphs of order a small number times a prime or a prime square

Journal of Combinatorial Theory, Series B

Kwak

2007

103

“…The first example of cubic one-regular graph was constructed by Frucht [5] with 432 vertices and lots of work have been done on cubic one-regular graphs as part of a more general problem dealing with the investigation of cubic arc-transitive graphs (see [6][7][8]). In 1997, Marušič and Xu [9] showed a way to construct a cubic one-regular graph Y from a tetravalent half-arc-transitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X. Thus, one can construct infinite many cubic one-regular graphs from the infinite family of tetravalent half-arc-transitive graphs with girth 3 constructed by Alspach et al in [10] and from another infinite family of tetravalent half-arc-transitive graphs with girth 3 constructed by Marušič and Nedela in [11].…”

Section: Introductionmentioning

confidence: 99%

Cubic one-regular graphs of order twice a square-free integer

Zhou

Sci. China Ser. A-Math.

2008

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. In this paper, we show that a cubic one-regular graph of order 2n exists if and only if n = 3 t p1p2 · · · ps 13, where t 1, s 1 and pi's are distinct primes such that 3| (pi − 1). For such an integer n, there are 2 s−1 non-isomorphic cubic one-regular graphs of order 2n, which are all Cayley graphs on the dihedral group of order 2n. As a result, no cubic one-regular graphs of order 4 times an odd square-free integer exist.

“…The first 1-regular cubic graph was constructed by Frucht [5] and later Miller [6] constructed infinitely many 1-regular cubic graphs of girth 6. Marušič and Xu [7] showed a way to construct a 1-regular cubic graph Y from a tetravalent halftransitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X, so that one may construct infinitely many 1-regular cubic graphs from the tetravalent half-transitive graphs of girth 3 given in refs. [8,9].…”

Section: Introductionmentioning

confidence: 99%

Classifying cubic symmetric graphs of order 10p or 10p 2

Kwak

2006

SCI CHINA SER A

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p 2 are also classified for each s 1 and each prime p, of which the proof depends on the classification of finite simple groups.