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On 2-arc-transitivity of Cayley graphs

Dragan Marušič
2003 Journal of combinatorial theory. Series B (Print)  
The classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra.  ...  Combin. 5 (1996) 83-86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups.  ...  In Section 2 we start by giving a brief history of the research on arc-transitive graphs and by laying out the basic strategy behind the analysis of 2-arc-transitive Cayley graphs of dihedral groups.  ... 
doi:10.1016/s0095-8956(02)00033-3 fatcat:vqaezpf3pffmzjfmph7oj3wfm4

On 2-arc-transitive Cayley graphs of Abelian groups

Primož Potočnik
2002 Discrete Mathematics  
A graph X is 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs of X . Some properties of 2-arc-transitive Cayley graphs of Abelian groups are considered.  ...  It is also proved that the set of generators of a 2-arc-transitive Cayley graph of an Abelian group which is not a circulant contains no elements of odd order.  ...  Some other examples of 2-arc-transitive Cayley graphs of Abelian groups were given in [2] where certain subclass of Cayley graphs of elementary Abelian groups were considered.  ... 
doi:10.1016/s0012-365x(01)00068-1 fatcat:nld4io737zfvnpkm5jjzd2dhqi

A Class of Arc-Transitive Cayley Graphs as Models for Interconnection Networks

Sanming Zhou
2009 SIAM Journal on Discrete Mathematics  
We study a class of Cayley graphs as models for interconnection networks.  ...  In particular, these hold for two important subclasses of graphs, namely, Cayley graphs admitting a complete rotation and Frobenius graphs of a certain type.  ...  Figure 2 : 2 Classes of Cayley graphs considered and their relations.  ... 
doi:10.1137/06067434x fatcat:5jxab4srovck5jtssetfanqb64

Arc-Transitive Cayley Graphs of Valency at Most Four on Abelian Groups

Jing Xu, Mingyao Xu
2001 Southeast Asian Bulletin of Mathematics  
In this paper we give a complete classi®cation for arc-transitive and one-regular Cayley graphs of valency at most four on ®nite abelian groups.  ...  Then all arc-transitive Cayley graphs of valency 3 and 4 on G are listed in Propositions 3.1 and 3.3.All one-regular Cayley graphs of valency 4 on G are listed in Proposition 3.3(2)(iv).The next theorem  ...  Let H hsiÂhti, then H q Z 2 2 . r The next proposition determines all normal, arc-transitive Cayley graphs of valency 3 and 4 on a non-cyclic abelian groups.  ... 
doi:10.1007/s10012-001-0355-z fatcat:ljpxbf5rxzfbjnogsiiksrkjqm

Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

Ali Reza Ashrafi, Bijan Soleimani
2018 Tamkang Journal of Mathematics  
In this paper, the connected Cayley graph of groups of order 3p q, for primes p and q, are investigated and all connected normal 1 2arc-transitive Cayley graphs of group of these orders will be classified  ...  The research of the authors are partially supported by the University of Kashan under grant no 159020/303.  ...  −arc transitive Cayley graph.  ... 
doi:10.5556/j.tkjm.49.2018.2169 fatcat:qi27ogdtefbqlde3ojdbdqfhqa

Corrigendum to "On 2-arc-transitivity of Cayley graphs" [J. Combin. Theory Ser. B 87 (2003) 162–196]

Dragan Marušič
2006 Journal of combinatorial theory. Series B (Print)  
A characterization of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups, is given in [3, Theorem 8.4], describing such graphs either explicitly or as cyclic covers of certain basic  ...  Consequently, the automorphism group A of a connected 2-arc-transitive Cayley graph X K 2n of a dihedral group D 2n = ρ, τ | ρ n = τ 2 = (ρτ ) 2 = 1 is necessarily imprimitive, allowing the above mentioned  ...  Let n 3, and let X be a connected, 2-arc-transitive Cayley graph of a dihedral group D = D 2n = ρ, τ | ρ n = τ 2 = (ρτ ) 2 = 1 of order 2n.  ... 
doi:10.1016/j.jctb.2006.01.003 fatcat:xc3pzitzlra7xgo4iapbz3q3na

On finite dual Cayley graphs

Jiangmin Pan
2020 Open Mathematics  
A Cayley graph \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of \Gamma (note that the right regular representation  ...  In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible.  ...  Funding: This work was partially supported by National Natural Science Foundation of China (11961076 and 11461007).  ... 
doi:10.1515/math-2020-0141 fatcat:auxp6aornzd4lk36hlrjawmcwq

Normal Edge-Transitive Cayley Graphs of the Group U6n

A. Assari, F. Sheikhmiri
2014 International Journal of Combinatorics  
A Cayley graph of a group G is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of  ...  In this paper, we determine all connected normal edge-transitive Cayley graphs of the group U6n.  ...  Aut(Γ) ( ( )) acts on the edges or arcs of Γ, respectively, and it is called normal halfarc-transitive Cayley graph if it is normal edge-transitive Cayley graph which is not normal arc-transitive.  ... 
doi:10.1155/2014/628214 fatcat:cys35jaiqrhufafoqhnxouaf6i

Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups [article]

Bo Ling, Ben Gong Lou
2017 arXiv   pre-print
To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.  ...  In this paper, we investigate the normality problem of the connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups.  ...  This completes the proof of the lemma.  ... 
arXiv:1702.05754v1 fatcat:ju7gzugukbe67ne4ju6e2anbmi

Cubic s-arc transitive Cayley graphs

Jing Jian Li, Zai Ping Lu
2009 Discrete Mathematics  
graph of G/ ( 3 ) 3 If N has at leat three orbits on V Γ, then Γ N is an X/N -edge-transitive Cayley graph of G/N and either (a) val(Γ N ) < val(Γ ) and Γ is not (X, 2)-arc-transitive; or .  ...  Then N has at least three orbits on V Γ and Γ is a cover of (the 'basic' (X/N, s)-arc-transitive Cayley graph) Γ N provided s ≥ 3.  ...  Construction of basic arc-transitive Cayley graph with given vertex stabilizer H and arc stabilizer P < H. Step1: Determine I = {τ ∈ N S n (P ) \ 1 =K¢H N S n (K)|τ 2 = 1, 1 τ = 1}.  ... 
doi:10.1016/j.disc.2009.05.002 fatcat:rjrvgwjyxvggtgeeyoxgm5r7n4

Vertex-transitive graphs that have no Hamilton decomposition [article]

Darryn Bryant, Matthew Dean
2014 arXiv   pre-print
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily  ...  The authors acknowledge the support of the Australian Research Council via grants DP120100790 and DP120103067, and are grateful to Marston Conder for helpful discussions.  ...  of a graph X and the vertices of K(L(X)), that Alt(5) has a regular action on the 2-arcs of F 10, and that PSL(2, 7) has a regular action on the 2-arcs of F 28, see [14] .  ... 
arXiv:1408.5211v3 fatcat:wrijoayk4bfwln472av3au73bq

On edge-transitive Cayley graphs of valency four

Xin Gui Fang, Cai Heng Li, Ming Yao Xu
2004 European journal of combinatorics (Print)  
Various new half-arc transitive graphs are constructed. A n−1 where n = 2 r 3 s−1 for r ∈ {2, 3, 4}  ...  A characterization is given of a class of edge-transitive Cayley graphs, providing methods for constructing Cayley graphs with certain symmetry properties.  ...  The third author is supported by Natural Science Foundation of China and the Research Fund for the Doctoral Program of Higher Education.  ... 
doi:10.1016/j.ejc.2003.07.008 fatcat:4imho3xnwjdnhdhpm4i7n3yg4y

Arc-transitive and s-regular Cayley graphs of valency 5 on Abelian groups

Mehdi Alaeiyan
2006 Discussiones Mathematicae Graph Theory  
A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs.  ...  In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups.  ...  The purpose of this paper is to investigate arc-transitive Cayley graphs of valency five on an Abelian group, that is, the arc-transitive graphs whose automorphism groups have an Abelian regular subgroup  ... 
doi:10.7151/dmgt.1328 fatcat:bqfooecg5vgqdjcac2bnzkemfe

Finite 2-arc-transitive abelian Cayley graphs

Cai Heng Li, Jiangmin Pan
2008 European journal of combinatorics (Print)  
Let Γ be a finite 2-arc-transitive Cayley graph of an abelian group.  ...  In particular, one of three cases occurs: Γ = K n,n − nK 2 where n is even but is not a 2-power, Γ has 2-power number of vertices, or Γ is a circulant.  ...  Acknowledgements This work was partially supported by an ARC Discovery Grant Project. The author is very grateful to the referees for their constructive comments.  ... 
doi:10.1016/j.ejc.2006.12.001 fatcat:whbecaouvfdp5ocbmp4s6jgv3m

Cubic core-free symmetric m-Cayley graphs

Jia-Li Du, Marston Conder, Yan-Quan Feng
2018 Journal of Algebraic Combinatorics  
This generalises the notions of a Cayley graph (where m = 1) and a bi-Cayley graph (where m = 2).  ...  We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31-41 2005).  ...  Cayley graphs admit a group of automorphisms that acts regularly (sharply transitively) on vertices.  ... 
doi:10.1007/s10801-018-0847-x fatcat:ffo63to5hzgqjle6nluv2v457u
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