Abstrakt
Is it true that every matching in the n-dimensional hypercube Q_n can be extended to a Gray code? More than two decades have passed since Ruskey and Savage asked this question and the problem still remains open. A solution is known only in some special cases, including perfect matchings or matchings of linear size.
This article shows that the answer to the Ruskey-Savage problem is affirmative for every matching of size at most n^2/16 + n/4. The proof is based on an inductive construction that extends balanced matchings in the completion of the hypercube K(Q_n) by edges of Q_n into a Hamilton cycle of K(Q_n).
On the other hand, we show that for every n >= 9 there is a balanced matching in K(Q_n) of size Theta(2^n/sqrt(n)) that cannot be extended in this way.