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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{dG(x),dG(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.
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