Finite 1-Regular Cayley Graphs of Valency 5
Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .
We assume that all graphs in this paper are finite, simple, and undirected.
Let be a graph. Denote the vertex set, arc set, and full automorphism group of by , , and , respectively. A graph is called -vertex-transitive or -arc-transitive if acts transitively on or , where . is simply called vertex-transitive, arc-transitive for the case where . In particular, is called -regular if acts regularly on its arcs and then 1-regular when .
Let be a finite group with identity element . For a subset of with , the Cayley graph of (with respect to ) is defined as the graph with vertex set such that are adjacent if and only if . It is easy to see that a Cayley graph has valency , and it is connected if and only if .
Li proved in  that there are only finite number of core-free -transitive Cayley graphs of valency for and and that, with the exceptions and , every -transitive Cayley graph is a normal cover of a core-free one. It was proved in  that there are core-free -transitive cubic Cayley graphs up to isomorphism, and there are no core-free -regular cubic Cayley graphs. A natural problem arises. Characterize -transitive Cayley graphs, in particular, which graphs are -regular? Until now, the result about -regular graphs mainly focused constructing examples. For example, Frucht gave the first example of cubic -regular graph in . After then, Conder and Praeger constructed two infinite families of cubic -regular graphs in . Marušič  and Malnič et al.  constructed two infinite families of tetravalent -regular graphs. Classifying such graphs has aroused great interest. Motivated by above results and problem, we consider -regular Cayley graphs of valency 5 in this paper.
A graph can be viewed as a Cayley graph of a group if and only if contains a subgroup that is isomorphic to and acts regularly on the vertex set. For convenience, we denote this regular subgroup still by . If contains a normal subgroup that is regular and isomorphic to , then is called an X-normal Cayley graph of ; if is not normal in but has a subgroup which is normal in and semiregular on with exactly two orbits, then is called an X-bi-normal Cayley graph; furthermore if , is called normal or bi-normal. Some characterization of normal and bi-normal Cayley graphs has given in [1, 2].
For a Cayley graph , is said to be core-free (with respect to ) if is core-free in some ; that is, .
The main result of this paper is the following assertion.
Theorem 1. Let be an -regular Cayley graph of valency , where . Let be the number of nonisomorphic core-free -regular Cayley graph of valency with the regular subgroup equal to . Then either (i)is an -normal or -bi-normal Cayley graph or(ii) is a nontrivial normal cover of one line of Table 1.In particular, there are no core-free -regular Cayley graphs of valency .
By Theorem 1, we can get the following remark immediately.
Remark 2. Let be an -regular Cayley graph of valency . Then is normal or bi-normal.
In this section we give some examples of graphs appearing in Theorem 1.
Example 3. Let be a cyclic group. Assume that is of order and . Let Suppose that where is an involution such that . Let be the Cayley graph of the dihedral group with respect to . Then is a connected -regular Cayley graph of valency . In particular, is -normal.
Noting , we may assume that . Since the involution is not equal to , we may let for some such that . Then , and so . Thus the element is of order as . So ; that is, is connected.
Obviously, and . However, ; then is an -regular normal Cayley graph of of valency .
Example 4. Let . Set and . Then and . Let such that . It follows that . Then for , and furthermore is -regular. Obviously is not normal in . However, is semiregular and has exactly two orbits on ; then is an -regular Cayley graph of valency . In particular, is -bi-normal.
3. The Proof of Main Results
In this section, we will prove our main results. We first present some properties about normal Cayley graphs.
For a Cayley graph , we have a subgroup of :
Clearly it is a subgroup of the stabilizer in of the vertex corresponding to the identity of . Since is connected, acts faithfully on . By Godsil [7, Lemma 2.1], the normalizer . So is a normal Cayley graph if and only if .
Let be an -regular Cayley graph of valency such that . Then contains at least one involution. Let , which is the core of in .
Lemma 5. Assume that . Then or .
Proof. Let be the stabilizer in of the vertex corresponding to the identity of . Then , , and . Let be the set of right cosets of in . Consider the action of on by the right multiplication. Then we get that is a primitive permutation group of degree and is a stabilizer of . Since has valency , , and so . Then we can show or , and then or , respectively.
Lemma 6. Suppose that and . Then is the icosahedron graph. Moreover, and is not -regular.
Proof. Note that , where , , and . Since has no nontrivial normal subgroup, is not bipartite. So is the icosahedron graph. Further by Magma , , so is not -regular.
Lemma 7. Suppose that and . Then the graph is not -regular and there is only one isomorphism class of these graphs.
Proof. Note that , , and . Let , where . By considering the right multiplication action of on the right cosets of in , can be viewed as a stabilizer of acting on . Without lost generality, we may assume that 1 is fixed by . Take an involution . Then, by , and we can identify with . Note that and ; then is one of the following: , , , , , , , and . Note . Assume that ; by calculation, we have , , , , and . Then , , , . Thus the corresponding is since for each . By similar argument, for every , we can work out explicitly, which is one of the following four cases: , , , and .
Now let . We declare that . Assume that . Note that ; then . Let . Since , and for any possible . Therefore , which leads to a contradiction. So the assertion is right; that is, is not -regular.
Let and for . Set , then and . It follows that and , namely, and . Now we consider . Note that and , then . Since , . On the other hand, and , then and . By the assumption, is not the graph satisfying conditions. So far we get the result that there is only one isomorphism class of graphs when .
To finish our proof, we need to introduce some definitions and properties. Assume that is an -vertex transitive graph with being a subgroup of . Let be a normal subgroup of . Denote the set of -orbits in by . The normal quotient of induced by is defined as the graph with vertex set , and two vertices , are adjacent if there exist and such that they are adjacent in . It is easy to show that acts transitively on the vertex set of . Assume further that is -edge-transitive. Then acts transitively on the edge set of , and the valency for some positive integer . If , then is called a normal cover of .
Proof of Theorem 1. Let be an -regular Cayley graph of valency , where . Then it is trivial to see that is connected. Let be the core of in . Assume that is not trivial. Then either or . The former implies ; that is, is an -normal Cayley graph with respect to . For the case where , it is easy to verify is an -bi-normal Cayley graph. Suppose that ; namely, has at least three orbits on . Since is a prime and is -regular, is a cover of and . We have that is a Cayley graph of and is core-free with respect to . Now suppose that is trivial, then is a core-free one. According to Lemmas 5, 6, and 7, there are two core-free -regular Cayley graphs of valency (up to isomorphism) as in Table 1. As far, Theorem 1 holds.
The project was sponsored by the Foundation of Guangxi University (no. XBZ110328), the Fund of Yunnan University (no. 2012CG015), NNSF (nos. 11126343, 11226141, and 11226045), and NSF of Guangxi (no. 2012GXNSFBA053010).