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Decomposition of multigraphs

100%
Kouider M., Mahéo M., Bryś K., Lonc Z.
Discussiones Mathematicae Graph Theory
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1998
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tom 18
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nr 2
225-232
EN
In this note, we consider the problem of existence of an edge-decomposition of a multigraph into isomorphic copies of 2-edge paths $K_{1,2}$. We find necessary and sufficient conditions for such a decomposition of a multigraph H to exist when (i) either H does not have incident multiple edges or (ii) multiplicities of the edges in H are not greater than two. In particular, we answer a problem stated by Z. Skupień.
2
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Decompositions of multigraphs into parts with two edges

80%
Ivančo J., Meszka M., Skupień Z.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 1
113-121
EN
Given a family 𝓕 of multigraphs without isolated vertices, a multigraph M is called 𝓕-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of 𝓕. We present necessary and sufficient conditions for the existence of such decompositions if 𝓕 comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.
3
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Decomposition of Complete Bipartite Multigraphs Into Paths and Cycles Having k Edges

80%
Jeevadoss S., Muthusamy A.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 4
715-731
EN
We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), [...] and k(p + q) = 2mn for k ≡ 0 (mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) =λ_mn, m, n ≥ k, (resp., m, n ≥ 3k/2) for k ≡ 0(mod 4) (respectively, for k ≡ 2(mod 4)). In fact, the necessary conditions given above are also sufficient when λ = 2.
4
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Decompositions into two paths

70%
Skupień Z.
Discussiones Mathematicae Graph Theory
|
2005
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tom 25
|
nr 3
325-329
EN
It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.
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