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Neighborhood conditions and edge-disjoint perfect matchings. (English) Zbl 0769.05073
A graph $$G$$ satisfies the all pairs neighborhood condition $$\text{ANC}(G)\geq m$$ if, for each pair $$x,y$$ of vertices of $$G$$, we have $$| N_ G(x)\cup N_ G(y)|\geq m$$. Let $$k$$ be a fixed positive integer and $$G$$ a graph of even order $$n$$ which satisfies the following conditions: (1) the minimum degree of $$G$$ is at least $$k+1$$; (2) the edge-connectivity of $$G$$ is at least $$k$$ and (3) $$\text{ANC}(G)\geq n/2$$. Then it is shown that for sufficiently large $$n$$, $$G$$ contains $$k$$ edge- disjoint perfect matchings. It is also shown that each of the conditions (1), (2) and (3) is necessary for $$G$$ to contain $$k$$ edge-disjoint perfect matchings.
##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C99 Graph theory
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