Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups
A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).
Let be a finite group, and let be a subset of such that . The Cayley graph = Cay of on is defined as the graph with a vertex set and edge set . Immediately from the definition there are three obvious facts (1) , the automorphism group of , contains the right regular representation of ; (2) is connected if and only if ; (3) is undirected if and only if .
A part of may be described in terms of automorphisms of , that is, the normalizer , a semidirect product of by , where =.
We simply use ) to denote the arc set of . A Cayley graph = Cay is said to be vertex transitive, edge transitive, and arc transitive if its automorphism group is transitive on the vertex set ), edge set ), and arc set ), respectively. For , an -arc in a graph is an ordered -tuple of vertices of such that is adjacent to for and for in other words, a directed walk of length which never includes a backtracking. A graph is said to be s arc transitive if is transitive on the set of -arcs in . In particular, 0 arc transitive means vertex transitive, and 1-arc transitive means arc transitive or symmetric. A subgroup of the automorphism group of a graph is said to be -regular if it acts regularly on the set of -arcs of .
It is difficult to find the full automorphism group of a graph in general, and so this makes it difficult to decide whether it is edge-transitive, even for a Cayley graph. As an accessible kind of edge transitive graphs, Praeger  focuses attention on those graphs for which is transitive on edges, and such a graph is said to be normal edge transitive. By the definition, every normal edge-transitive Cayley graph is edge-transitive, but not every edge-transitive Cayley graph is normal edge-transitive.
Independently for our investigation, and as another attempt to study the structure of finite Cayley graphs, Xu  defined a Cayley graph = Cay to be normal if ) is normal subgroup of the full automorphism group . Xu's concept of normality for a Cayley graph is a very strong condition. For example, is normal if and only if . However any edge-transitive Cayley graph which is normal, in the sense of Xu's definition, is automatically normal edge transitive.
Praeger posed the following question in : what can be said about the structure of Cayley graphs which are edge transitive but not normal edge transitive? In , Alaeiyan et al. have given partial answer to this question for abelian groups of valency at most 5, and also Sim and Kim  determined normal edge-transitive circulant graphs. In the next theorem we will identify all cubic edge transitive Cayley graphs of dihedral group which are not normal edge-transitive. This is a partial answer to Question 5 of . Throughout of this paper, we suppose that , and = Cay is connected and undirected cubic Cayley graph. The main result of this paper is the following theorem.
Theorem 1.1. Let be a dihedral group, and let = Cay(G,S) be a connected cubic Cayley graph. If is an edge-transitive Cayley graph but is not normal edge transitive, then , satisfy one of the following: (1), , ;(2), , , the generalized Peterson graph.
2. Basic Facts
In this section we give some facts on Cayley graphs, which will be useful for our purpose. First we make some comments about the normalizer of the regular subgroup . As before, the normalizer of the regular subgroup in the symmetric group is the holomorph of , that is, the semidirect product . Thus,
The following lemmas are basic for our purpose. Now we have the first lemma from .
Lemma 2.1 (see [1, Proposition 1]). Let = Cay(G,S) be a Cayley graph for a finite group . Then is normal edge transitive if and only if is either transitive on or has two orbits in which are inverse of each other.
Lemma 2.2 (see [2, Proposition 1.5]). Let = Cay(G,S), and . Then is normal if and only if , where is the stabilizer of 1 in .
Lemma 2.3 (see [5, Lemma 4.4]). All 1-regular cubic Cayley graphs on the dihedral group are normal.
Lemma 2.4 (see [6, Lemma 3.2]). Let be a connected cubic graph on dihedral group , and let and be two orbits of . Also let be the subgroup of fixing setwise and , respectively. If acts unfaithfully on one of and , then .
Let be the core of in . By assuming the hypothesis in the above lemma, we have the following results
Lemma 2.5 (see [6, Lemma 3.5]). If is a proper subgroup of , then is isomorphic to or .
Lemma 2.6 (see [6, Lemma 3.6]). If , then is isomorphic to , where (mod ), and .
Let . Then the elements of are and , where . All are involutions, and is an involution if and only if is even and . Finally in this section we obtain a preliminary result restricting for cubic Cayley graphs of Cay(. We can easily prove the following lemma
Lemma 2.7. Let = Cay be Cayley graphs of . Then is cubic, connected and, undirected if and only if one of the following conditions holds (1)When is odd, one has(2)When is even, one has
Let and be two graphs. The direct product is defined as the graph with vertex set such that for any two vertices and in , is an edge in whenever and or and . Two graphs are called relatively prime if they have no nontrivial common direct factor. The lexicographic product is defined as the graph with vertex set such that for any two vertices and in , is an edge in whenever or and . Let . Then there is a natural embedding in , where for , the th copy of is the subgraph induced on the vertex subset in . The deleted lexicographic product is the graph obtained by deleting all the edges (natural embedding) of from .
3. Proof of Theorem 1.1
As we have seen in Section 1, each edge transitive Cayley graph which is normal is automatically normal edge transitive. Hence for the proof of Theorem 1.1, we must determine all nonnormal connected undirected cubic Cayley graphs for dihedral group . If , then dihedral group is isomorphic to , and so it is easy to show that the cubic Cayley graph Cay is normal. So from now we assume that . Also, since Cay when is disconnected, thus we do not consider this case for the proof of the main theorem. First we prove the following lemma.
Lemma 3.1. Let be the dihedral group with , and let = Cay (G,S) be a cubic Cayley graph. Then (a)if is and is connected, then holds;(b)if is or and is connected, then if . For , and is or , one has; and is connected and normal;(c)if is or , then always holds.
Proof. (a) Suppose first that . Then
We show that . Suppose to the contrary that . We may suppose that . Now we have , where . Hence is not connected, which is a contradiction.
(b) Now suppose that or , that is, . For , we have . We claim that . Suppose to the contrary that . We may suppose that . Then , where Cay and . So is not connected, which is a contradiction. Now let . Then , , or , respectively. Therefore , , or , respectively. Obviously is connected, and . Also we have , and Cay Cay Cay. Let be an automorphism of Cay, which fixes 1 and all elements of . Since , and , we have and . Therefore , and , and hence fixes all elements of . Thus , and acts faithfully on . So we may view as a permutation group on . Now let be an arbitrary element of . Since , we have . If or , then or , which is a contradiction. Thus , and is generated by the permutation . So . On the other hand, is an element of . Therefore , and hence by Lemma 2.2, is normal.
(c) Finally, suppose that or , that is, . Then . Clearly . The results now follow.
By considering this lemma, we prove the following proposition. This result will be used in the proof of Theorem 1.1.
Proposition 3.2. Let be the dihedral group , and let be a connected and undirected cubic Cayley graph. Then is normal except y one of the following cases happens: (1);(2) (the generalized Peterson graph);(3);(4) (Heawood's graph).
Proof. First assume that . Since is connected, by Lemma 3.1(a), . Now consider the graph , and let be an automorphism of Cay, which fixes 1 and all elements of . Since , , and , we have ,, and , respectively. Therefore , , and , and hence fixes all elements of . Because of the connectivity of , this automorphism is the identity in . Therefore acts faithfully on . So we may view as a permutation group on . Now let be an arbitrary element of . Since , we have . If , or , then , or , respectively. Now again we consider . In this subgraph, and have valency 2, and , have valency 1. This implies a contradiction. Thus , and is generated by the permutation . So . On the other hand, is an element of . Therefore , and hence by Lemma 2.2, is normal.
Now assume that , or . If , then by Lemma 3.1 (b), Cay, and is normal. Now if , then again by Lemma 3.1(b), . Considering the graph , with the same reason as before if an automorphism of fixes 1 and all elements of , then it also fixes all elements of . Because of the connectivity of , this automorphism is the identity in . Therefore acts faithfully on . So we may view as a permutation group on . We can easily see that is generated by the permutation . So . On the other hand, is an element of . Therefore , and hence by Lemma 2.2, is normal.
Finally assume that , or . Up to graph isomorphism, , where . In this case, is a bipartite graph with the partition , where and are just two orbits of , and we assume the block contains 1. Let be the subgroup of fixing setwise and , respectively. If acts unfaithfully on one of and , then by Lemma 2.4, , and is not in but in , and so is not normal. Let act faithfully on and . Then . If , then is isomorphic to , and is not in but in , and so is not normal. From now on we assume . Now suppose that , the core of in , is a proper subgroup of . Then by Lemma 2.5, Cay or Cay). For the first case, is not in but in , and so is not normal. For the second case, is not in but in , and so is not normal. Finally we suppose that . Then by Lemma 2.6, is isomorphic to the Cay, where (mod and . The Cayley graph is 1-regular, and by Lemma 2.3, is normal. The result now follows.
Now we complete the proof of Theorem 1.1. We remind that any edge transitive Cayley graph which is normal, in the sense of Xu's definition, is also normal edge transitive. Thus this implies that we must consider nonnormal Cayley graphs which were obtained in Proposition 3.2. So we consider four cases in Proposition 3.2. For case (1), we claim that there is no automorphism of such that maps to . Suppose to the contrary that there is an automorphism such that maps to . Then must be mapped to , where , and so with the simple check it is easy to see that this is a contradiction. Also in case (2), with the same reason as above there is a contradiction. Hence does not act transitively on also does not have two orbits in which are inverse of each other. Now by using Lemma 2.1 these graphs are not normal edge transitive. For the last two cases it is easy to show that acts transitively on , and hence by Lemma 2.1, these graphs are normal edge transitive. Now the proof is complete as claimed.
Y. Q. Feng, J. H. Kwak, and M. Y. Xu, “s-regular cubic Cayley graphs on abelian or dihedral groups,” Research Report 53, Institute of Mathematics Peking University, 2000.View at: Google Scholar
S. F. Du, Y. Q. Feng, J. H. Kwak, and M. Y. Xu, “Cubic Cayley graphs on dihedral groups,” Mathematical Analysis and Applications, vol. 7, pp. 224–234, 2004.View at: Google Scholar