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Decomposition of bipartite graphs into closed trails. (English) Zbl 1224.05402
Summary: Let \(\operatorname {Lct}(G)\) denote the set of all lengths of closed trails that exist in an even graph \(G\). A sequence \((t_1,\dots ,t_p)\) of elements of \(\operatorname {Lct}(G)\) adding up to \(| E(G)| \) is \(G\)-realisable provided there is a sequence \((T_1,\dots ,T_p)\) of pairwise edge-disjoint closed trails in \(G\) such that \(T_i\) is of length \(t_i\) for \(i=1,\dots ,p.\) The graph \(G\) is arbitrarily decomposable into closed trails if all possible sequences are \(G\)-realisable. In the paper, it is proved that, if \(a\geq 1\) is an odd integer and \(M_{a,a}\) is a perfect matching in \(K_{a,a}\), then the graph \(K_{a,a}-M_{a,a}\) is arbitrarily decomposable into closed trails.

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
Full Text: DOI EuDML
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