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Note on the decomposition of $$\lambda K_{m,n}$$ ($$\lambda K^*_{m,n}$$) into paths. (English) Zbl 0578.05054
Let $$G$$ and $$H$$ be (di)graphs. By an $$H$$-decomposition of $$G$$ we mean a partition of the edge set of $$G$$ into disjoint subsets each spanning in $$G$$ a subgraph isomorphic to $$H$$. In the paper $$H$$-decompositions of a complete bipartite symmetric multidigraph $$\lambda K^*_{m,n}$$ and a complete bipartite multigraph $$\lambda K_{m,n}$$ are studied in the case when $$H$$ is a path of length $$k$$. The following three results are proved:
(1) Let $$m\geq n$$. $$\lambda K^*_{m,n}$$ has a decomposition into paths of length $$k$$ if and only if $$k$$ divides $$2\lambda mn$$, $$m\geq \lceil (k+1)/2\rceil$$ and $$n\geq \lceil k/2\rceil$$.
(2) Let $$\lambda$$ be an even integer and let $$m\geq n$$. $$\lambda K_{m,n}$$ has a decomposition into paths of length $$k$$ if and only if $$k$$ divides $$\lambda mn$$, $$m\geq \lceil (k+1)/2\rceil$$ and $$n\geq \lceil k/2\rceil$$.
(3) Let $$m$$ and $$n$$ be even, $$m\geq n$$. $$\lambda K_{m,n}$$ has a decomposition into paths of length $$k$$ if and only if $$k$$ divides $$\lambda mn$$, $$m\geq \lceil (k+1)/2\rceil$$ and $$n\geq \lceil k/2\rceil$$. Some results in the case when at least one of $$m$$ and $$n$$ is odd are also presented.
Reviewer: M. Truszczyński

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C38 Paths and cycles 05C20 Directed graphs (digraphs), tournaments
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##### References:
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