Pseudo-cartesian product and hamiltonian decompositions of Cayley graphs on abelian groups

Fan, Cong; Lick, Don R.; Liu, Jiuqiang

doi:10.1016/0012-365x(95)00035-u

Cited by 13 publications

(20 citation statements)

References 11 publications

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“…. , i + k is isomorphic to a Cayley graph on an abelian group as observed in [10]. The subgraph Y is not bipartite so that it is Hamilton-connected by Theorem 1.7.…”

Section: Basic Toolkitmentioning

confidence: 99%

Hamilton paths in Cayley graphs on generalized dihedral groups

2010

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“…. , i + k is isomorphic to a Cayley graph on an abelian group as observed in [10]. The subgraph Y is not bipartite so that it is Hamilton-connected by Theorem 1.7.…”

Section: Basic Toolkitmentioning

confidence: 99%

Hamilton paths in Cayley graphs on generalized dihedral groups

2010

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“…By Theorem 1.1 X is HD. The following theorem, proved in [12], forms the basis for a portion of the proof of the Main Result. Theorem 3.3 ([12]).…”

Section: Preliminariesmentioning

confidence: 99%

“…This section lists some techniques to create Hamilton cycles, relying considerably on the notation and techniques developed in [14] and [12]. Let A n = a 1 a 2 · · · a n a 1 and B m = b 1 b 2 · · · b m b 1 denote cycles of length n and m. .…”

Section: Color-switching Configurationsmentioning

confidence: 99%

“…Alspach's conjecture remains open in general (see the survey by Curran-Gallian [9]) despite being investigated from many vantage points: the valency, the group order, the group type, restrictions on the connection set, etc. The following theorem is a summary of complete results when only the valency is considered: Theorem 1.1 (Alspach et al [3], Bermond et al [5], Dean [11], Fan et al [12], Liu [14,15], Westlund et al [19]). Every connected k-regular Cayley graph on a finite Abelian group A is Hamilton-decomposable if k ≤ 5 or k = 6 and |A| is odd.…”

Section: Introductionmentioning

confidence: 99%

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Hamilton decompositions of 6-regular Cayley graphs on even Abelian groups with involution-free connections sets

Westlund

2014

Discrete Mathematics

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Abstract. Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamiltondecomposable. Liu has shown that for |A| even, if S = {s 1 , . . . , s k } ⊂ A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A; S ), is decomposable into k Hamilton cycles, where S denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regular case, this article relaxes the constraint of strong minimality on S to require only that S be strongly a-minimal, for some a ∈ S and the index of a be at least four. Strong a-minimality means that 2s / ∈ a for all s ∈ S \ {a, −a}. Some infinite families of open cases for the 6-regular Cayley graphs on even order Abelian groups are resolved. In particular, if |s 1 | ≥ |s 2 | > 2|s 3 |, then Cay(A; {s 1 , s 2 , s 3 } ) is Hamilton-decomposable.

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“…The case = 3 is still open, although many partial results exist, see [8,9,[22][23][24]. There are also results for > 3, see [2,12,[17][18][19]. It was shown in [4] that there exist 2 -valent connected Cayley graphs on finite non-abelian groups that are not Hamilton-decomposable.…”

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confidence: 99%

On Hamilton decompositions of infinite circulant graphs

Bryant

Herke

Maenhaut

et al. 2017

Journal of Graph Theory

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The natural infinite analog of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2 -valent infinite circulant graph has a two-wayinfinite Hamilton path, there exist many such graphs that do not have a decomposition into edge-disjoint twoway-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2 -valent connected circulant graph has a decomposition into edge-disjoint Hamilton cycles. We settle the problem of decomposing 2 -valent infinite circulant graphs into edge-disjoint twoway-infinite Hamilton paths for = 2, in many cases when = 3, and in many other cases including where the connection set is ±{1, 2, … , } or ±{1, 2, … , − 1, + 1}. K E Y W O R D Scirculant graph, hamilton decomposition, infinite graph

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Pseudo-cartesian product and hamiltonian decompositions of Cayley graphs on abelian groups

Cited by 13 publications

References 11 publications

Hamilton paths in Cayley graphs on generalized dihedral groups

Hamilton paths in Cayley graphs on generalized dihedral groups

Hamilton decompositions of 6-regular Cayley graphs on even Abelian groups with involution-free connections sets

On Hamilton decompositions of infinite circulant graphs

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