Hamiltonian decompositions of complete regular s-partite graphs

Hilton, A. J. W.; Rodger, C. A.

doi:10.1016/0012-365x(86)90186-x

Cited by 46 publications

(42 citation statements)

References 6 publications

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“…Since each layer is either connected, or is a subgraph of a connected subgraph of G Gð2Þ, and since each layer is connected to the layer above and below, it follows that G Gð2Þ is connected. Therefore the graph G colored with c c satifies properties (1)(2)(3)(4)(5), so the result follows from Theorem 2.1. …”

Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 62%

“…Now suppose that mð p À 1Þ is even. If q ¼ 1, then s 1 ¼ mp and the problem reduces to finding a hamilton decomposition of K ðpÞ m ; this has already been solved (see [3,4]). For the remainder of the proof, we assume that q !…”

Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 96%

“…By Theorem 2.1, it suffices to construct an edge-colored graph G that satisfies conditions (1)(2)(3)(4)(5). G is constructed by first amalgamating K, an edge-colored graph isomorphic to K We begin with the edge-coloring c : EðKÞ !…”

Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 99%

“…Laskar and Auerbach [4] showed in 1976 that the complete p-partite graph K a 0 ;...;a pÀ1 has a hamilton decomposition if and only if both mð p À 1Þ is even and m ¼ a 0 ¼ Á Á Á ¼ a pÀ1 . Using the technique of vertex amalgamations (or graph homomorphisms), Hilton and Rodger [3] presented new proofs of both results. The methods developed by Hilton and Rodger proved to be very powerful in finding hamilton decompositions of other graphs.…”

Section: Introductionmentioning

confidence: 97%

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Hamilton decompositions of complete multipartite graphs with any 2‐factor leave

Leach

Rodger

2003

Journal of Graph Theory

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For m ! 1 and p ! 2, given a set of integers s 1 ,. . .,s q with s j ! p þ 1 for 1 j q and P q j¼1 s j ¼ mp, necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph K m ,. . ., m ÀEðUÞ, where U is a 2-factor of K m ,. . ., m consisting of q cycles, the jth cycle having length s j . This result is then used to completely solve the problem when p ¼ 3, removing the condition that s j ! p þ 1. ß

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Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 62%

Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 96%

Section: Hamilton Decompositions With a 2-regular Leavementioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Hamilton decompositions of complete multipartite graphs with any 2‐factor leave

Leach

Rodger

2003

Journal of Graph Theory

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“…[14][15][16]21] for example) to prove that for every 2-regular graph F 0 of odd order n, there exists a 2-factorization of K n in which one 2-factor is isomorphic to F 0 and the remaining 2-factors are Hamilton cycles. Here we extend this result by completely settling the 2-factorization problem for K n in the case where three of the 2-factors are isomorphic to arbitrary given 2-regular graphs, and the remaining 2-factors are Hamilton cycles (see Theorem 3.1).…”

Section: Introductionmentioning

confidence: 98%

Hamilton cycle rich two‐factorizations of complete graphs

Bryant

2004

J of Combinatorial Designs

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For all integers n ! 5, it is shown that the graph obtained from the n-cycle by joining vertices at distance 2 has a 2-factorization is which one 2-factor is a Hamilton cycle, and the other is isomorphic to any given 2-regular graph of order n. This result is used to prove several results on 2-factorizations of the complete graph K n of order n. For example, it is shown that for all odd n ! 11, K n has a 2-factorization in which three of the 2-factors are isomorphic to any three given 2-regular graphs of order n, and the remaining 2-factors are Hamilton cycles. For any two given 2-regular graphs of even order n, the corresponding result is proved for the graph K n À I obtained from the complete graph by removing the edges of a 1-factor. #

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Detachments of Amalgamated 3‐Uniform Hypergraphs Factorization Consequences

Bahmanian¹

2012

J of Combinatorial Designs

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Abstract. A detachment of a hypergraph F is a hypergraph obtained from F by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph G can be thought of as taking G , partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph F . In this paper we use Nash-Williams lemma on laminar families to prove a detachment theorem for amalgamated 3-uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger and Nash-Williams.To demonstrate the power of our detachment theorem, we show that the complete 3-uniform n-partite multi-hypergraph λK 3 m1,...,mn can be expressed as the union G 1 Y . . . Y G k of k edge-disjoint factors, where for i " 1, . . . , k, G i is r i -regular, if and only if: (i) m i " m j :" m for all 1 ď i, j ď k, (ii) 3 r i mn for each i, 1 ď i ď k, and (iii) ř k i"1 r i " λ`n´1 2˘m 2 .

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Hamiltonian decompositions of complete regular s-partite graphs

Cited by 46 publications

References 6 publications

Hamilton decompositions of complete multipartite graphs with any 2‐factor leave

Hamilton decompositions of complete multipartite graphs with any 2‐factor leave

Hamilton cycle rich two‐factorizations of complete graphs

Detachments of Amalgamated 3‐Uniform Hypergraphs Factorization Consequences

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