Bipartite graphs with every k-matching in a Hamiltonian cycle

Let G = (U, W; E) be a balanced bipartite graph of order 2n. We call G k-matching-hamiltonian, if every matching with k edges is contained in a hamiltonian cycle. In this paper, we prove that the property that being k-matching-hamiltonian is (n + k + 1)-stable, that is, whenever G + xy is k-matching-hamiltonian for some x E U, y E W such that xy (is an element of) over bar E(G) and d(x) + d(y) >= n + k + 1 then G is k-matching-hamiltonian. Thus we give a sufficient condition for a balanced bipartite graph to be k-matching-hamiltonian and hamiltonian.