Degree condition for the existence of a k-factor containing a given Hamiltonian cycle

Let k be an integer with k>=2 and let G be a graph having sufficiently large order n. Suppose that kn is even, the minimum degree of G is at least k and max{d"G(x),d"G(y)}>=(n+@a)/2 for each pair of nonadjacent vertices x and y in G, where @a=3 for odd k and @a=4 for even k. Then G has a k-factor (i.e. a k-regular spanning subgraph) which contains a given Hamiltonian cycle C if G-E(C) is connected.