Decomposing Complete Equipartite Multigraphs into Cycles of Variable Lengths: The Amalgamation‐detachment Approach

Bahmanian, Amin; Šajna, Mateja

doi:10.1002/jcd.21419

Cited by 10 publications

(32 citation statements)

References 26 publications

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“…This result was recently generalised to complete multigraphs [6]. Partial results have also been obtained for decompositions of complete bipartite graphs [9,12,19] and complete multipartite graphs [1]. Here, we add to this body of work by addressing the question of when a complete graph with a hole admits a decomposition into cycles of arbitrary specified lengths.…”

Section: Introductionmentioning

confidence: 89%

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DecomposingKu+w−Kuinto cycles of prescribed lengths

Horsley

Hoyte

2017

Discrete Mathematics

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a b s t r a c tWe prove that the complete graph with a hole K u+w − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most min(u, w), and the longest cycle is at most three times as long as the second longest. This generalises existing results on decomposing the complete graph with a hole into cycles of uniform length, and complements work on decomposing complete graphs, complete multigraphs, and complete multipartite graphs into cycles of arbitrary specified lengths.

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Section: Introductionmentioning

confidence: 89%

“…It is routine to check using (v) that some pair in the appropriate row will always satisfy these conditions. Case (a 3 , c 3 ) (1,2) This choice ensures that a 2 , a 3 , c 2 and c 3 are nonnegative integers such that a 2 is even, a 2 + a 3 = a, c 2 + c 3 = c,…”

Section: Lemma 28 Let U ≥ 5 and W ≥ 8 Be Integers Such That U Is Oddmentioning

confidence: 98%

DecomposingKu+w−Kuinto cycles of prescribed lengths

Horsley

Hoyte

2017

Discrete Mathematics

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“…A C α 4 , C β 5 -decomposition of K m • K n was given by Fu [22]. Moreover, Bahmanian andŠajna [7] showed that if K m (λn) has a decomposition into cycles of lengths k 1 , k 2 , . .…”

Section: And the Directed Hamilton Cycle Decompositions Of The Symmetmentioning

confidence: 99%

Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

Paulraja¹,

Srimathi²

2021

Discuss. Math. Graph Theory

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By a C α 3 , C β 6-decomposition of a graph G, we mean a partition of the edge set of G into α cycles of length 3 and β cycles of length 6. In this paper, necessary and sufficient conditions for the existence of a C α 3 , C β 6decomposition of (K m × K n)(λ), where × denotes the tensor product of graphs and λ is the multiplicity of the edges, is obtained. In fact, we prove that for λ ≥ 1, m, n ≥ 3 and (m, n) = (3, 3), a C α 3 , C β 6-decomposition of (K m × K n)(λ) exists if and only if λ(m − 1)(n − 1) ≡ 0 (mod 2) and 3α + 6β = λm(m−1)n(n−1) 2 .

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“…In an earlier paper [6], the authors of the present paper developed a technique for constructing a

(c_{1} m, …, c_{k} m)

‐cycle decomposition of

λ K_{n \times m}

from a

(c_{1}, …, c_{k})

‐cycle decomposition of

λ m K_{n}

. This method, detachment, was applied to the decompositions guaranteed by [10], resulting in many new cycle decompositions of complete equipartite multigraphs.…”

Section: Introductionmentioning

confidence: 99%

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

Bahmanian

Šajna

2021

J of Combinatorial Designs

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We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both complete multigraphs and complete equipartite multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions.

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Decomposing Complete Equipartite Multigraphs into Cycles of Variable Lengths: The Amalgamation‐detachment Approach

Cited by 10 publications

References 26 publications

DecomposingKu+w−Kuinto cycles of prescribed lengths

DecomposingKu+w−Kuinto cycles of prescribed lengths

Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

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