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Given that Q_d is the hypercube graph of dimension d then it is a known fact (not so trivial to prove though) that given a perfect matching M of Q_d (d >= 2) it is possible to find another perfect matching N of Q_d such that M \cup N is a Hamiltonian cycle in Q_d.

The question now is - given a (non necessarily perfect) matching M of Q_d (d >= 2) is it possible to find a set of edges N such that M \cup N is a Hamiltonian cycle in Q_d.

The statement is proven to be true for d in {2,3,4}

edited Nov 17 2009 at 14:49

Greg Kuperberg
36.3k●2●116●179

asked Nov 13 2009 at 13:06

Jernej
842●5●18

aorq · Accepted Answer · 2009-11-13 20:27:34Z UTC

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This is a known open problem. See "Matchings extend to Hamiltonian cycles in hypercubes" over at the Open Problem Garden.

answered Nov 13 2009 at 20:27

aorq
2,470●17●36

Is every matching of the hypercube graph extensible to a Hamiltonian cycle

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