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On edge-Hamiltonian property of Cayley graphs

1988
*
Discrete Mathematics
*

Hence every

doi:10.1016/0012-365x(88)90191-4
fatcat:h2mvnqlj4ncwre3s56jvnonyve
*Cayley**graph**of*order a power*of*2 is*edge*-*Hamiltonian*. ... A*Cayley**graph*ouer G is defined as a*graph*G(X) whose vertex set is G and whose*edge*set consists*of*all unordered pairs [a, b] with a, b E G and am'b E X U X-', where X-t denotes the set (x-t ( .x E ...*Edge**Hamiltonian**property**of**Cayley**graphs*With the basic lemmas established in the previous section, we are now in a position to prove the following theorems. Theorem 5. ...##
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On edge-hamiltonian Cayley graphs

1994
*
Discrete Applied Mathematics
*

Chen (1988) conjectured that every finite

doi:10.1016/0166-218x(94)90091-4
fatcat:4uxp7nbk7zbi3dwhxmqcdtcqxq
*hamiltonian**Cayley**graph*is*edge*-*hamiltonian*. We prove some*hamiltonian**Cayley**graphs*to be*edge*-*hamiltonian*and some*Cayley**graphs*to be*hamiltonian*. ... Results*of*Alspach and Zhang (1989)*on*Hamilton cycles in*Cayley**graphs**on*dihedral groups are generalized. 0166-218X/94/$07.00 0 199LElsevier Science B.V. ...*Edge*-*hamiltonian**property**of**Cayley**graphs*In investigating whether a given*hamiltonian**Cayley**graph*has the*property*that each*of*its*edges*lies*on*a Hamilton cycle, the following result*of*Chen can be ...##
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Hamiltonian decomposition of Cayley graphs of degree 4

1989
*
Journal of combinatorial theory. Series B (Print)
*

We prove that any 4-regular connected

doi:10.1016/0095-8956(89)90040-3
fatcat:huxfuaizhrdojbly4new3v6xre
*Cayley**graph**on*a finite abelian group can be decomposed into two*hamiltonian*cycles. ... This answers a partial case*of*Alspach's conjecture concerning*hamiltonian*decompositions*of*2k-regular connected*Cayley**graphs*. ... Such a*property*is interesting because routings*on*a*hamiltonian*cycle are easy; these*graphs*are good fault-tolerant networks because in case*of**edge*failure*one*can always find a simple routing. ...##
###
An evidence for Lov´asz conjecture about Hamiltonian paths and cycles

2010
*
Matemática contemporânea
*

In this work, we show some

doi:10.21711/231766362010/rmc3914
fatcat:as4bgm3waba35eon5x332oafva
*properties**of*the gadget*graph*H l,p which was used by Holyer to prove the NP−completeness*of*the problem*of*the*edge*-partition into cliques [3] . ... However, since none*of*these four*graphs*is a*Cayley**graph*, we can look at the Lovász conjecture as stating that every connected*Cayley**graph*with more than two vertices has a*Hamiltonian*cycle. ... The*graph*H l,p has the following*properties*: 1. the degree*of*each vertex is 2( l 2 ); the number*of*vertices is p l−1 ; 3. the number*of**edges*is ( l 2 )p l−1 . ...##
###
Page 3572 of Mathematical Reviews Vol. , Issue 91G
[page]

1991
*
Mathematical Reviews
*

A

*graph*G whose square G* is*Hamiltonian*is called a critical*graph*if the removal*of*any*edge**of*G results in a*graph*whose square is not*Hamiltonian*. ... In this paper, the authors give a sufficient condition for*Cayley*digraphs to have*Hamiltonian*circuits, and give a lower bound*on*the number*of**Hamiltonian*circuits in a*Hamiltonian**Cayley*digraph. ...##
###
Hamiltonian Decomposition of Recursive Circulants
[chapter]

1998
*
Lecture Notes in Computer Science
*

The result is not only a partial answer to the problem posed by Alspach that every connected

doi:10.1007/3-540-49381-6_32
fatcat:c7dww2hcyfh4dkip4zj3sq25q4
*Cayley**graph*over an abelian group is*hamiltonian*decomposable, but also an extension*of*Micheneau's that recursive ... We show that recursive circulant G(cd m ; d) is*hamiltonian*decomposable. Recursive circulant is a*graph*proposed for an interconnection structure*of*multicomputer networks in 8]. ...*Graph*classes*Hamiltonian*decomposition is*one**of*the most interesting strong hamiltonicity, that is, a*hamiltonian**property*which implies the existence*of*a*hamiltonian*cycle. ...##
###
Approximating Cayley diagrams versus Cayley graphs
[article]

2011
*
arXiv
*
pre-print

We construct a sequence

arXiv:1103.4968v3
fatcat:q5djorgymreo3fvojrpanbffq4
*of*finite*graphs*that weakly converge to a*Cayley**graph*, but there is no labelling*of*the*edges*that would converge to the corresponding*Cayley*diagram. ... We give an example where this subtree is a*Hamiltonian*cycle, but convergence is meant in a stronger sense. These latter are related to whether having a*Hamiltonian*cycle is a testable*graph**property*. ... The other sequence, K n will also be the direct product*of*a*graph*B n and C 4 , and it will have the*property*that it contains a*Hamiltonian*cycle such that every other*edge**of*the*Hamiltonian*cycle is ...##
###
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. ...##
###
Embeddings on Torus-Butterfly Interconnection Network

2012
*
International Journal of Applied Information Systems
*

From the

doi:10.5120/ijais12-450817
fatcat:l3cwgx65fbf2xecu7whq4w3eou
*properties**of**Cayley**graphs*which have*Hamiltonian*path, the linear array and 2D-Mesh can be embedded into this new Torus-Butterfly network with minimum dilation and expansion. ... Torus-Butterfly network is a*Cayley**graph*. ... As already mentioned in the introduction,*one**of*the models that are widely used as interconnection networks are*Cayley**graphs*, this is caused by the*Cayley**graph*has the*properties*finite, connect, undirected ...##
###
Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups
[article]

2020
*
arXiv
*
pre-print

A well-known conjecture

arXiv:2006.09759v1
fatcat:dok7hkzr4vewppoduc36rscuam
*of*Alspach says that every 2k-regular*Cayley**graph**of*an abelian group can be decomposed into*Hamiltonian*cycles. ... In this setting*one*natural analogue*of*a*Hamiltonian*cycle is a spanning double-ray. ... Let G be a locally finite*graph*. Then a set*of**edges*C is a circle if and only if C meets every finite cut F*of*G in an even number*of**edges*, and there is no non-empty C ′ C with this*property*. ...##
###
Page 3282 of Mathematical Reviews Vol. , Issue 85h
[page]

1985
*
Mathematical Reviews
*

*one*

*edge*between them, then G is

*Hamiltonian*. ... We thereby make use

*of*the algebraic

*properties*

*of*G to establish that at least

*one*

*of*a pair

*of*cycles can be lifted appropriately.” Pareek, C. M. (KWT-KUWA); Skupien, Z. ...

##
###
Some problems on Cayley graphs

2008
*
Linear Algebra and its Applications
*

This survey paper presents the historical development

doi:10.1016/j.laa.2008.05.010
fatcat:aw4jerefane77kfhanjxqeivvi
*of*some problems*on**Cayley**graphs*which are interesting to*graph*and group theorists such as Hamiltonicity or diameter problems, to computer scientists ... Acknowledgments I thank Reza Khosrovshahi for inviting me to participate at the First IPM (Institute for Studies in Theoretical Physics and Mathematics) conference*on*Algebraic*Graph*Theory that was held ...*on*April 21-26, 2007, in Tehran and for his hospitality during my staying in Iran. ...##
###
Page 2608 of Mathematical Reviews Vol. , Issue 96e
[page]

1996
*
Mathematical Reviews
*

)
The

*Hamiltonian**property**of**Cayley**graphs**on*symmetric groups. ... Sci. 11 (1994), no. 3, 16-18. 05 COMBINATORICS 2608 96e:05074b 05C25 05C45 20B30 Wang, Shi Ying [Wang, Shi Ying”] (PRC-XINJ-EM; Urumgi) The*Hamiltonian**property**of**Cayley**graphs**on*symmetric groups. ...##
###
Biswapped Networks and Their Topological Properties

2007
*
Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing (SNPD 2007)
*

We show how key parameters

doi:10.1109/snpd.2007.217
dblp:conf/snpd/XiaoCHWP07
fatcat:o4pdvrfxb5b4lenkdmddq6ytii
*of*a BSN are related to the corresponding parameters*of*its basis network and obtain a number*of*results*on*internode distances,*Hamiltonian*cycles, and node-disjoint paths. ... In particular, if the basis network is a*Cayley*digraph then so is the resulting BSN. ... Acknowledgment Research*of*the first four authors was supported by the Natural Science Foundation*of*China and Guangdong Province (No.04020130). ...##
###
Hamiltonian paths in Cayley graphs

2009
*
Discrete Mathematics
*

In Section 4 we try to emphasize the group theoretic and algebraic combinatorial

doi:10.1016/j.disc.2009.02.018
fatcat:efhn3mkayzgl3hsvpljbcyoe3y
*properties**of**Cayley**graphs*. ... Here we present a construction*of**Hamiltonian*3-regular*Cayley**graphs*, and prove that these are expanders. ... We shall ignore labels and orientation*of**edges*and treat Γ as a simple*graph**on*|G| vertices. Clearly, Γ is d-regular, where d = |S|. From this point*on*, we consider only*Cayley**graphs*. ...
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