Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups

For a group G, and a subset S of G such that 1 G ∈ S, let Γ = Cay(G, S) be the corresponding Cayley graph. Then Γ is said to be normal edge transitive, if N Aut(Γ) (G) is transitive on edges. In this paper we determine all connected, undirected edge-transitive Cayley graphs of finite abelian groups with valency at most five, which are not normal edge transitive. This is a partial answer to a question of Praeger.

Discrete Mathematics, 1993

Sanders, R.S., Graphs on which a dihedral group acts edge-transitively, Discrete Mathematics 118 (1993) 225-232.

2021

A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...

Bulletin of The Australian Mathematical Society, 2021

A graph Γ is called (G, s)-arc-transitive if G ≤ Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s + 1 vertices (v 0 , v 1 ,. .. , v s) of Γ such that v i−1 and v i are adjacent for 1 ≤ i ≤ s and v i−1 = v i+1 for 1 ≤ i ≤ s− 1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL 2 (11), M 11 , M 23 and A 11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M 11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

Proyecciones (Antofagasta), 2021

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

Journal of Graph Theory, 2012

In 1983, the second author [D. Marušič, Ars Combinatoria 16B (1983), 297-302] asked for which positive integers n there exists a non-Cayley vertex-transitive graph on n vertices. (The term non-Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265-269] asked to determine the smallest valency ϑ(n) among valencies of non-Cayley vertex-transitive graphs of order n. As cycles are clearly Cayley graphs, ϑ(n) ≥ 3 for any non-Cayley number n. In this paper a goal is set to determine those non-Cayley numbers n for which ϑ(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non-Cayley vertex-transitive Contract grant sponsors: Agencija za raziskovalno dejavnost Republike Slovenije, research program P1-0285 (to K. K., D. M., C. Z.); Agencija za raziskovalno dejavnost Republike Slovenije, proj. mladi raziskovalci (to C. Z.).

2021

For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transiti...

Journal of Graph Theory, 1996

The Petersen graph on 10 vertices is the smallest example of a vertextransitive graph which is not a Cayley graph. In 1983, D. Marušič asked: for what values of n does there exist such a graph on n vertices? We give several new constructions of families of vertex-transitive graphs which are not Cayley graphs and complete the proof that, if n is divisible by p 2 for some prime p, then there is a vertex-transitive graph on n vertices which is not a Cayley graph unless n is p 2 , p 3 , or 12.

Journal of Algebraic Combinatorics, 2013

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T , S). We first prove that if Cay(T , S) is neither cyclic nor K 4[2] , then a \ {1} S for all a ∈ S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

Journal of Algebraic Combinatorics, 2015

A Cayley Graph for a group G is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of G (the normaliser of a regular copy of G in Sym(G)). We complete the classification of normal edge-transitive Cayley graphs of order a product of two primes by dealing with Cayley graphs for Frobenius groups of such orders. We determine the automorphism groups of these graphs, proving in particular that there is a unique vertex-primitive example, namely the flag graph of the Fano plane.