Hamiltonian paths with prescribed edges in hypercubes Original Research Article

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