Geodesic
Decompositions of multigraphs into parts with the same size
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 335-347

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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 existence of such decompositions if ℱ consists of all multigraphs of size q except for one. Namely, for a multigraph H of size q we find each multigraph M of size kq, such that every partition of the edge set of M into parts of cardinality q contains a part which induces a submultigraph of M isomorphic to H.
Keywords: edge decompositions, multigraphs 
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     author = {Ivanco, Jaroslav},
     title = {Decompositions of multigraphs into parts with the same size},
     journal = {Discussiones Mathematicae. Graph Theory},
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     url = {http://geodesic.mathdoc.fr/item/DMGT_2010_30_2_a12/}
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Ivanco, Jaroslav. Decompositions of multigraphs into parts with the same size. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 335-347. http://geodesic.mathdoc.fr/item/DMGT_2010_30_2_a12/