Hamiltonian paths with prescribed edges in hypercubes

Given a set P of at most 2n − 4 prescribed edges (n ≥ 5) and vertices u and v whose mutual distance is odd, the n-dimensional hypercube Qn contains a hamiltonian path between u and v passing through all edges of P iff the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as endvertices. This resolves a problem of Caha and Koubek who showed that for any n ≥ 3 there exist vertices u, v and 2n−3 edges of Qn not contained in any hamiltonian path between u and v, but still satisfying the condition above. The proof of the main theorem is based on an inductive construction whose basis for n = 5 was verified by a computer search. Classical results on hamiltonian edge-fault tolerance of hypercubes are obtained as a corollary.

• Type: Journal
• Year: 2007
• Where: DM
• Authors: Tomás Dvorák, Petr Gregor
• Keywords: DM 2007, Hamiltonian Path, Mutual Distance, N-dimensional Hypercube Qn