Degree condition for the existence of a -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 k n is even, the minimum degree of G is at least k and max { d G ( x ) , d G ( y ) } ≥ ( n + α ) / 2 for each pair of nonadjacent vertices x and y in G , where α = 3 for odd k and α = 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.