Hamiltonian paths in hypercubes with local traps

Abstract

The n-dimensional hypercube Qn is a graph with 2n vertices, each labeled with a distinct binary string of length n. The vertices are connected by an edge if and only if their labels differ in one bit. The hypercube is bipartite, the set of nodes is the union of two sets: nodes of parity 0 (the number of ones in their labels is even) and nodes of parity 1 (the number of ones is odd). We consider Hamiltonian paths in hypercubes with faulty edges and prove the following: (1) If Qn has one vertex u of degree 1, then u can be connected by a Hamiltonian path with every vertex v that is of a parity different than u and that is not connected with u by a healthy edge. (2) If Qn with n ≥ 4 has two vertices u and v of degree 1, then they can be connected by a Hamiltonian path if the distance between u and v is odd and greater than 1 or if u and v are connected by the faulty edge. (3) If Qn contains a cycle (u, v, w, x) in which all edges going away from the cycle from u and w are faulty, then u or w can be connected by a Hamiltonian path with any vertex outside the cycle that is of different parity than u and w.Moreover, in all three cases, the thesis remains true even if Qn has n−3 additional faulty edges. Furthermore, in all three cases, no other Hamiltonian paths are possible.