Research Article  Open Access
Jing Jian Li, Ben Gong Lou, Xiao Jun Zhang, "Finite 1Regular Cayley Graphs of Valency 5", International Journal of Combinatorics, vol. 2013, Article ID 125916, 3 pages, 2013. https://doi.org/10.1155/2013/125916
Finite 1Regular Cayley Graphs of Valency 5
Abstract
Let and . We say is regular Cayley graph if acts regularly on its arcs. is said to be corefree if is corefree 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 corefree ones up to isomorphism. In particular, there are no corefree regular Cayley graphs of valency .
1. Introduction
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 vertextransitive or arctransitive if acts transitively on or , where . is simply called vertextransitive, arctransitive for the case where . In particular, is called regular if acts regularly on its arcs and then 1regular 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 [1] that there are only finite number of corefree transitive Cayley graphs of valency for and and that, with the exceptions and , every transitive Cayley graph is a normal cover of a corefree one. It was proved in [2] that there are corefree transitive cubic Cayley graphs up to isomorphism, and there are no corefree 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 [3]. After then, Conder and Praeger constructed two infinite families of cubic regular graphs in [4]. Marušič [5] and Malnič et al. [6] 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 Xnormal 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 Xbinormal Cayley graph; furthermore if , is called normal or binormal. Some characterization of normal and binormal Cayley graphs has given in [1, 2].
For a Cayley graph , is said to be corefree (with respect to ) if is corefree 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 corefree regular Cayley graph of valency with the regular subgroup equal to . Then either (i)is an normal or binormal Cayley graph or(ii) is a nontrivial normal cover of one line of Table 1.In particular, there are no corefree 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 binormal.
2. Examples
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.
Proof. Let
where .
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 binormal.
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 [8], , 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 [2], 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 edgetransitive. 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 binormal 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 corefree with respect to . Now suppose that is trivial, then is a corefree one. According to Lemmas 5, 6, and 7, there are two corefree regular Cayley graphs of valency (up to isomorphism) as in Table 1. As far, Theorem 1 holds.
Acknowledgments
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).
References
 C. H. Li, “Finite $s$arc transitive Cayley graphs and flagtransitive projective planes,” Proceedings of the American Mathematical Society, vol. 133, no. 1, pp. 31–41, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 J. J. Li and Z. P. Lu, “Cubic $s$arc transitive Cayley graphs,” Discrete Mathematics, vol. 309, no. 20, pp. 6014–6025, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 R. Frucht, “A oneregular graph of degree three,” Canadian Journal of Mathematics, vol. 4, pp. 240–247, 1952. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 M. D. E. Conder and C. E. Praeger, “Remarks on pathtransitivity in finite graphs,” European Journal of Combinatorics, vol. 17, no. 4, pp. 371–378, 1996. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Marušič, “A family of oneregular graphs of valency 4,” European Journal of Combinatorics, vol. 18, no. 1, pp. 59–64, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Malnič, D. Marušič, and N. Seifter, “Constructing infinite oneregular graphs,” European Journal of Combinatorics, vol. 20, no. 8, pp. 845–853, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. D. Godsil, “On the full automorphism group of a graph,” Combinatorica, vol. 1, no. 3, pp. 243–256, 1981. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system. I. The user language,” Journal of Symbolic Computation, vol. 24, no. 34, pp. 235–265, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2013 Jing Jian Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.