Matchings extend to Hamiltonian cycles in hypercubes with faulty edges

Abstract

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube Qn; and obtain the following results. Let n ⩾ 3; M ⊂ E(Qn); and F ⊂ E(Qn)M with 1 ⩽ |F| ⩽ 2n − 4 − |M|: If M is a matching and every vertex is incident with at least two edges in the graph Qn − F; then all edges of M lie on a Hamiltonian cycle in Qn − F: Moreover, if |M| = 1 or |M| = 2; then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for |M| = 1