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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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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##### References:
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