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Decomposition of complete bipartite multigraphs into paths and cycles having k edges
Discussiones Mathematicae Graph Theory
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 > k, and k(p + q) = 2mn for k ≡ 0 (mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0 (mod 2), k(p + q) = λmn, 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.doi:10.7151/dmgt.1830 fatcat:eio7rr6wfrchbfmp53ebl6ga4i