Papers › Matchings in Hypercubes Extend to Long Cycles
Matchings in Hypercubes Extend to Long Cycles
Jiří Fink, Vojtěch Hotmar
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The n-dimensional hypercube graph Qₙ has as vertices all subsets of {1, …, n}, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the n-dimensional hypercube Qₙ can be extended into a Hamilton cycle. We prove that matchings of Qₙ containing edges spanning at most d = 5 directions can be extended into a Hamilton cycle. We also characterize when these matchings of most d = 5 directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary d and n where d ≤n assuming some extension properties hold in Q_d which we verified by a computer for d=5.
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