Geodesic
Decomposition of Complete Bipartite Multigraphs Into Paths and Cycles Having $k$ Edges
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 715-731

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We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph K_m,n ( λ ) into paths and cycles having k edges. In particular, we show that such decomposition exists in K_m,n ( λ ), when λ≡ 0 (mod 2), m,n ≥ k/2, m+n gt; k and k(p + q) = 2mn for k ≡ 0 (mod 2) and also when λ≥ 3, λ m ≡λ n ≡ 0(mod 2), k(p + q) =λ m n, 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.
Keywords: path, cycle, graph decomposition, multigraph 
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Jeevadoss, Shanmugasundaram; Muthusamy, Appu. Decomposition of Complete Bipartite Multigraphs Into Paths and Cycles Having $k$ Edges. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 715-731. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a9/