Degree conditions for the existence of [k, k+1]-factors containing a given Hamiltonian cycle

Let k ≥ 2 be an integer and G a 2-connected graph of order |G| ≥ 3 with minimum degree at least k. Suppose that |G| ≥ 8k − 16 for even |G| and |G| ≥ 6k − 13 for odd |G|. We prove that G has a [k, k + 1]-factor containing a given Hamiltonian cycle if max{degG(x), degG(y)} ≥ |G|/2 for each pair of nonadjacent vertices x and y in G. This is best possible in the sense that there exists a graph having no k-factor containing a given Hamiltonian cycle under the same conditions. The lower bound of |G| is also sharp.