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Hamilton decompositions of complete multipartite graphs with any 2-factor leave. (English) Zbl 1031.05108
Summary: For \(m \geq 1\) and \(p \geq 2\), given a set of integers \(s_1,\dots ,s_q\) with \(s_j \geq p+1\) for \(1 \leq j \leq q\) and \(\Sigma_{j=1}^q s_j=mp\), necessary and sufficient conditions are found for the existence of a Hamilton decomposition of the complete \(p\)-partite graph \(K_{m,\dots ,m}-E(U)\), where \(U\) is a 2-factor of \(K_{m,\dots ,m}\) consisting of \(q\) cycles, the \(j\)th cycle having length \(s_j\). This result is then used to completely solve the problem when \(p = 3\), removing the condition that \(s_j \geq p+1\).

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C45 Eulerian and Hamiltonian graphs
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