On pancyclism in hamiltonian graphs

Kouider, Mekkia; Marczyk, Antoni

doi:10.1016/s0012-365x(01)00333-8

Cited by 11 publications

(6 citation statements)

References 6 publications

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“…In [185] we constructed a hamiltonian graph G of order n with two vertices x and y satisfying d(x) + d(y) ≥ 2n − 4r − 2 (3 ≤ 3r < n − 1) that contains no cycle of length n − r + 1. In the light of the two examples we conjecture the following.…”

Section: Case 22: P < α(S)mentioning

confidence: 99%

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Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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Section: Case 22: P < α(S)mentioning

confidence: 99%

“…The following theorem of Kouider and Marczyk [185] is another generalization of Proposition 4.5 due to Bondy [55].…”

Section: Case 22: P < α(S)mentioning

confidence: 99%

Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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“…The study of Pancyclicity of networks has gained much attention in the literature [11][12][13][14][15][16][17]. Bondy [11] has proved that if the minimum degree of a network of size N is N /2, then it is a pancyclic graph.…”

Section: Pancyclicity Of Pyramidsmentioning

confidence: 99%

“…Pancyclicity, Hamiltonian-connectedness and the possession of link-disjoint Hamiltonian cycles, are three important properties of network topologies that have been studied vastly in the past [1,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The pancyclicity of a network represents its capacity for cycles of different lengths to be embedded in it.…”

Section: Introductionmentioning

confidence: 99%

The Grid-Pyramid: A Generalized Pyramid Network

Farahabady¹,

Sarbazi-Azad

2006

J Supercomput

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The Pyramid network is a desirable network topology used as both software data-structure and hardware architecture. In this paper, we propose a general definition for a class of pyramid networks that are based on grid connections between the nodes in each level. Contrary to the conventional pyramid network in which the nodes in each level form a mesh, the connections between these nodes may also be according to other grid-based topologies such as the torus, hypermesh or WK-recursive. Such pyramid networks form a wide class of interconnection networks that possess rich topological properties. We study a number of important properties of these topologies for general-purpose parallel processing applications. In particular, we prove that such pyramids are Hamiltonian-connected, i.e. for any arbitrary pair of nodes in the network there exists at least one Hamiltonian path between the two given nodes, and pancyclic, i.e. any cycle of length 3, 4 . . . and N , can be embedded in a given N -node pyramid network. It is also proven that two link-disjoint Hamiltonian cycles exist in the torus-pyramid and hypermesh-pyramid networks.

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“…The pancycle problem involves finding all possible lengths of cycles in a graph G. More precisely, letting V(G) and jV(G)j be the set and the number of vertices of G, respectively, the goal is to embed a cycle of length l into G for every l satisfying b 6 l 6 jV(G)j, where b is a specific positive integer. This problem has attracted a great deal of mathematicians [4,5,21,24] since it was brought up by Bondy [5] in 1971. Recently, many researchers studied this problem in the area of interconnection networks [2,7,11,12,14,17,27].…”

Section: Introductionmentioning

confidence: 99%