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hypercube; perfect matching; 1-factor; $r$-extendability
A graph having a perfect matching is called $r$-extendable if every matching of size $r$ can be extended to a perfect matching. It is proved that in the hypercube $Q_n$, a matching $S$ with $ |S|\leq n$ can be extended to a perfect matching if and only if it does not saturate the neighbourhood of any unsaturated vertex. In particular, $Q_n$ is $r$-extendable for every $r$ with $1\leq r\leq n-1.$
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