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Vertex-primitive digraphs of prime-power order are hamiltonian

Ming-Yao Xu
1994 Discrete Mathematics  
In this note we generalize Witte's result to connected vertex-primitive digraphs of prime-power order; namely, we prove that every connected vertex-primitive digraph of prime-power order is hamiltonian  ...  Witte [6] proved that every connected Cayley digraph of a p-group is hamiltonian. It is reasonable to attempt to generalize this result to vertex-transitive digraphs of prime-power order.  ...  Vertex-primitive digraphs of prime-power order are hamiltonian. Proof.  ... 
doi:10.1016/0012-365x(94)90134-1 fatcat:wadxguyznzcxxig7bcskwbgs34

Hamiltonian cycles and paths in Cayley graphs and digraphs — A survey

Stephen J. Curran, Joseph A. Gallian
1996 Discrete Mathematics  
Since the 1984 survey of results on hamiltonian cycles and paths in Cayley graphs by Witte and Gallian, many advances have been made.  ...  Moreover, Babai has shown that all graphs can be realized as an induced subgraph of a Cayley graph of any sufficiently large group.  ...  Every Cayley graph on a group G = (a>H of dihedral type where H has prime-power order or [H I = pqr where p, q and r are (not necessarily distinct) primes is hamiltonian.  ... 
doi:10.1016/0012-365x(95)00072-5 fatcat:wlr3et6cdvhediytnvdz2tnetm

2-generated Cayley digraphs on nilpotent groups have hamiltonian paths [article]

Dave Witte Morris
2011 arXiv   pre-print
We show that if a,b is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path.  ...  This implies there is a hamiltonian path in every connected Cayley graph on G that has valence at most 4.  ...  If G = P × A, where P has prime-power order, and A is abelian, then every connected Cayley digraph on G has a hamiltonian path. Remark.  ... 
arXiv:1103.5293v3 fatcat:ezylptlxhze4tm7k4lu5eu22pi

A survey: Hamiltonian cycles in Cayley graphs

David Witte, Joseph A. Gallian
1984 Discrete Mathematics  
It has been conjectured there is a hamiltonian cycle in every Cayley graph. Interest in this and other closely related questions has grown in the past few years. We su~-;ey the ~PCU!!  ...  There is a hamiltonian c*cuit in every Cayley digraph on a group -of. prime-power order whose commutator s;ubgroup is cyclic (W&e [33D, Let us sketch theargument~ for abelian groups of prime-power order  ...  Suppose the commutator subgroup of G is a cyclic group of prime-power order. Then there is a hamiltonian cycle in every Cayley graph on G. Ides of P-f. Let N be the coznnutator subgroup of G.  ... 
doi:10.1016/0012-365x(84)90010-4 fatcat:vgd2fxdwgnhkvnorhzlkbh2g6m

On Cayley Digraphs That Do Not Have Hamiltonian Paths

Dave Witte Morris
2013 International Journal of Combinatorics  
We construct an infinite family {Cay→(Gi;ai;bi)} of connected, 2-generated Cayley digraphs that do not have hamiltonian paths, such that the orders of the generators ai and bi are unbounded.  ...  We also prove that if G is any finite group with |[G,G]|≤3, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]|=4 or 5).  ...  The other results in this paper International Journal of Combinatorics 7 were obtained during a visit to the School of Mathematics and Statistics at the University of Western Australia (partially supported  ... 
doi:10.1155/2013/725809 fatcat:eekbhjaiozg4fiwo4qjfa42h2q

The isomorphism problem for Cayley digraphs on groups of prime-squared order

Anne Joseph
1995 Discrete Mathematics  
Witte and Gallian [ 13] provide a survey of results of Hamiltonian cycles in Cayley graphs. Given any prime p, there are two groups of order p2.  ...  Klin and P6schel [6] solved the isomorphism problem for Cayley digraphs on cyclic groups of prime-power order. Their proof makes heavy use of Schur rings.  ...  Proof of Main Theorem Proof o[1 ~ 2 Let X = Cay(G : T) be a Cayley digraph on a group G of order p2, where p is prime. Assume X is isomorphic to a Cayley digraph on both 7/~ and 7/p x 7/p.  ... 
doi:10.1016/0012-365x(93)e0215-p fatcat:k32jmynylfabjok6e4it5wptvq

Page 3163 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
A new infinite family of vertex-transitive graphs that are not Cayley graphs is constructed as follows. Let G be a finite or infinite group. Let p be an odd prime.  ...  Second, he proves that if S = {s;, 52,53} is a minimal generating set of the finite abelian group G, S has either two elements of order 2 or one element of prime order, and X(G;S) has even degree, then  ... 

Contents

2005 Discrete Mathematics  
Brunat On endo-Cayley digraphs: The hamiltonian property 194 D.W. Morris, J. Morris and D.P. Moulton Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups 208 J.  ...  Dobson On groups of odd prime-power degree that contain a full cycle 65 E. Dobson and J. Morris On automorphism groups of circulant digraphs of square-free order 79 D. Froncek and B.  ... 
doi:10.1016/s0012-365x(05)00407-3 fatcat:rxvtxpybhzgy3krwwreznodnvy

On Cayley digraphs that do not have hamiltonian paths [article]

Dave Witte Morris
2013 arXiv   pre-print
We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large.  ...  We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).  ...  The other results in this paper were obtained during a visit to the School of Mathematics and Statistics at the University of Western Australia (partially supported by funds from Australian Research Council  ... 
arXiv:1306.5443v1 fatcat:7xau6xkgq5enbdxgea6feybefy

On Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order [article]

Edward Dobson, Ademir Hujdurović, Klavdija Kutnar, Joy Morris
2015 arXiv   pre-print
Morris have shown that every non-CCA group G contains a section isomorphic to the nonabelian group F_21 of order 21. We first show that there is a unique non-CCA Cayley graph Γ of F_21.  ...  We then show that if Cay(G,S) is a non-CCA graph of a group G of odd square-free order, then G = H× F_21 for some CCA group H, and Cay(G,S) = Cay(G,T)Γ.  ...  Introduction and preliminaries We consider Cayley digraphs Cay(G, S) of a group G with connection set S whose arcs (g, gs) are colored with the color s, for s ∈ S.  ... 
arXiv:1512.00239v1 fatcat:l6bxz4gp5jf73hlqjgeu6uhade

On the structure of Hamiltonian cycles in Cayley graphs of finite quotients of the modular group

Paul E. Schupp
1998 Theoretical Computer Science  
It is a fairly longstanding conjecture that if G is any finite group with IG/ > 2 and if X is any set of generators of G then the Cayley graph T(G : X) should have a Hamiltonian cycle.  ...  It turns out that in the case where G is a finite quotient of the modular group the Hamiltonian cycles possess remarkable structural properties.  ...  I am particularly grateful to Nigel Boston for his patience in showing me how to use the computer algebra system CAYLEY and its successor MAGMA.  ... 
doi:10.1016/s0304-3975(98)00041-3 fatcat:732oc4jdpzfzxjtib56e4kpj34

Hamilton cycles and paths in vertex-transitive graphs—Current directions

Klavdija Kutnar, Dragan Marušič
2009 Discrete Mathematics  
In this article current directions in solving Lovász's problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given.  ...  could prove that for a prime p, a connected regular Z p -cover of a hamiltonian vertex-transitive graph of order a power of p, is hamiltonian.  ...  This, being a connected vertex-transitive graph of prime power order smaller than p k , may be assumed to be hamiltonian.  ... 
doi:10.1016/j.disc.2009.02.017 fatcat:urzirc6srzgrjfbxtyus4tsshy

Brian Alspach and his work

Joy Morris, Mateja Šajna
2005 Discrete Mathematics  
His home is often opened to visitors who are at various stages of their careers, but particularly to students and to mathematicians who are still establishing themselves.  ...  Brian maintains a keen interest in applications of mathematics, particularly those in industry, which he feels are an important part of maintaining young adults' interest in mathematics.  ...  Any remaining mistakes are entirely our own.  ... 
doi:10.1016/j.disc.2005.03.024 fatcat:umr2plo3hbb3jbinmsmxoewg3u

Page 624 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
For any prime p, it is shown that every vertex transitive digraph of order p*, k < 3, is a Cayley digraph. (This generalizes a theorem of J. Turner [J. Combin.  ...  Keating and the reviewer [this col- lection, see heading, 89-102] have further extended this to groups whose commutator subgroup is cyclic of prime-power order.  ... 

Page 4978 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
B 40 (1986), no. 1, 107- 112] has proved that a connected Cayley digraph on p* vertices, p a prime and p* # 2, has a Hamilton circuit.  ...  power number of vertices.  ... 
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