Hamiltonian paths with prescribed edges in hypercubes

Abstract

Given a set P of at most 2 n - 4 prescribed edges ( n ⩾ 5 ) and vertices u and v whose mutual distance is odd, the n-dimensional hypercube Q n 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 2 n - 3 edges of Q n 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.