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Vertex-primitive digraphs of prime-power order are hamiltonian

1994
*
Discrete Mathematics
*

In this note we generalize Witte's result to connected vertex-primitive

doi:10.1016/0012-365x(94)90134-1
fatcat:wadxguyznzcxxig7bcskwbgs34
*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. ...##
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Hamiltonian cycles and paths in Cayley graphs and digraphs — A survey

1996
*
Discrete Mathematics
*

Since the 1984 survey

doi:10.1016/0012-365x(95)00072-5
fatcat:wlr3et6cdvhediytnvdz2tnetm
*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*. ...##
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2-generated Cayley digraphs on nilpotent groups have hamiltonian paths
[article]

2011
*
arXiv
*
pre-print

We show that if a,b is any 2-element generating set

arXiv:1103.5293v3
fatcat:ezylptlxhze4tm7k4lu5eu22pi
*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. ...##
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A survey: Hamiltonian cycles in Cayley graphs

1984
*
Discrete Mathematics
*

It has been conjectured there is a

doi:10.1016/0012-365x(84)90010-4
fatcat:vgd2fxdwgnhkvnorhzlkbh2g6m
*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. ...##
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On Cayley Digraphs That Do Not Have Hamiltonian Paths

2013
*
International Journal of Combinatorics
*

We construct an infinite family {Cay→(Gi;ai;bi)}

doi:10.1155/2013/725809
fatcat:eekbhjaiozg4fiwo4qjfa42h2q
*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 ...##
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The isomorphism problem for Cayley digraphs on groups of prime-squared order

1995
*
Discrete Mathematics
*

Witte and Gallian [ 13] provide a survey

doi:10.1016/0012-365x(93)e0215-p
fatcat:k32jmynylfabjok6e4it5wptvq
*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. ...##
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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 ...##
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Contents

2005
*
Discrete Mathematics
*

Brunat
On endo-

doi:10.1016/s0012-365x(05)00407-3
fatcat:rxvtxpybhzgy3krwwreznodnvy
*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. ...##
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On Cayley digraphs that do not have hamiltonian paths
[article]

2013
*
arXiv
*
pre-print

We construct an infinite family

arXiv:1306.5443v1
fatcat:7xau6xkgq5enbdxgea6feybefy
*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 ...##
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On Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order
[article]

2015
*
arXiv
*
pre-print

Morris have shown that every non-CCA group G contains a section isomorphic to the nonabelian group F_21

arXiv:1512.00239v1
fatcat:l6bxz4gp5jf73hlqjgeu6uhade
*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. ...##
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On the structure of Hamiltonian cycles in Cayley graphs of finite quotients of the modular group

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

doi:10.1016/s0304-3975(98)00041-3
fatcat:732oc4jdpzfzxjtib56e4kpj34
*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. ...##
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Hamilton cycles and paths in vertex-transitive graphs—Current directions

2009
*
Discrete Mathematics
*

In this article current directions in solving Lovász's problem about the existence

doi:10.1016/j.disc.2009.02.017
fatcat:urzirc6srzgrjfbxtyus4tsshy
*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*. ...##
###
Brian Alspach and his work

2005
*
Discrete Mathematics
*

His home is often opened to visitors who

doi:10.1016/j.disc.2005.03.024
fatcat:umr2plo3hbb3jbinmsmxoewg3u
*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. ...##
###
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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