Hamiltonian cycles and paths with prescribed edges in hypercubes

The n-dimensional hypercube Qn is a graph whose vertex set consists of all binary vectors of length n, two vertices being adjacent whenever the corresponding vectors differ in exactly one coordinate. It is well-known that Qn is hamiltonian for every n ≥ 2 and the research on hamiltonian cycles satisfying certain additional properties has received a considerable attention ([7]). The applications in parallel computing inspired study of hypercubes with faulty links, which lead to the investigation of hamiltonian cycles of Qn avoiding certain set of forbidden edges ([2, 3, 8]). A problem in a sense complementary has been proposed recently by R. Caha and V. Koubek: Given a set of prescribed edges in a hypercube, under which conditions there always exists a hamiltonian cycle, passing through every edge of this set? In [1] they observed that any proper subset P of edges of a hamiltonian cycle necessarily induces a subgraph consisting of pairwise vertex-disjoint paths and showed that in case |P| ≤ n− 1, n ≥ 2, this condition is also sufficient to guarantee the existence of a hamiltonian cycle of Qn, passing through every edge of P. On the other hand, for any n ≥ 3 there is a set of 2n− 2 edges, satisfying the above condition, but not contained in any hamiltonian cycle. Indeed, let v be an arbitrary vertex of Qn and P a set of edges incident with neighbors of v so that each but one neighbor of v is incident with exactly two edges of P, but no edge of P is incident with v. It is not difficult to see that this can be always done in such a way that the above condition is preserved. Since Qn is a regular graph of degree n, it follows that |P| = 2n−2 and apparently, any cycle passing through all edges of P avoids v. The main purpose of this contribution is to show that the upper bound obtained in this way is really sharp: