Toughness for the existence of k-Hamiltonian [a, b]-factors

Let a, b and k be three nonnegative integers with a >= 2 and b >= a(k + 1) + 2. A graph G is called a k-Hamiltonian graph if G - U contains a Hamiltonian cycle for every subset U subset of V (G) with vertical bar U vertical bar = k. An [a, b]-factor F of G is called a Hamiltonian [a, b] factor if F contains a Hamiltonian cycle. If G - U has a Hamiltonian [a, b]-factor for every subset U subset of V(G) with vertical bar U vertical bar = k, then we say that G admits a k-Hamiltonian [a, b] -factor. Suppose that G is a k Hamiltonian graph of order n with n >= a + k + 2. In this paper, it is proved that G includes a k-Hamiltonian [a, b]-factor if delta(G) >= a + k and t(G) >= a - 1 + (a - 1)(k + 1)/b - 2.