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Hence every 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. ...doi:10.1016/0012-365x(88)90191-4 fatcat:h2mvnqlj4ncwre3s56jvnonyve
Chen (1988) conjectured that every finite 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 ...doi:10.1016/0166-218x(94)90091-4 fatcat:4uxp7nbk7zbi3dwhxmqcdtcqxq
We prove that any 4-regular connected 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. ...doi:10.1016/0095-8956(89)90040-3 fatcat:huxfuaizhrdojbly4new3v6xre
In this work, we show some 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  . ... 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 . ...doi:10.21711/231766362010/rmc3914 fatcat:as4bgm3waba35eon5x332oafva
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. ...
Lecture Notes in Computer Science
The result is not only a partial answer to the problem posed by Alspach that every connected 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. ...doi:10.1007/3-540-49381-6_32 fatcat:c7dww2hcyfh4dkip4zj3sq25q4
We construct a sequence 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 ...arXiv:1103.4968v3 fatcat:q5djorgymreo3fvojrpanbffq4
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
International Journal of Applied Information Systems
From the 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 ...doi:10.5120/ijais12-450817 fatcat:l3cwgx65fbf2xecu7whq4w3eou
A well-known conjecture 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. ...arXiv:2006.09759v1 fatcat:dok7hkzr4vewppoduc36rscuam
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. ...
This survey paper presents the historical development 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. ...doi:10.1016/j.laa.2008.05.010 fatcat:aw4jerefane77kfhanjxqeivvi
) 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. ...
We show how key parameters 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). ...doi:10.1109/snpd.2007.217 dblp:conf/snpd/XiaoCHWP07 fatcat:o4pdvrfxb5b4lenkdmddq6ytii
In Section 4 we try to emphasize the group theoretic and algebraic combinatorial 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. ...doi:10.1016/j.disc.2009.02.018 fatcat:efhn3mkayzgl3hsvpljbcyoe3y
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