Toughness and the existence of Hamiltonian [a, b]-factors of graphs

The toughness of a graph G, denoted by t(G), is defined as t(G) = min{vertical bar S vertical bar/omega(G - S) : S subset of V (G), omega(G - S) >= 2} where omega(G - S) denotes the number of components of G - S or t(G) = +infinity if G is a complete graph. Let a and b be nonnegative integers with 2 <= a < b - 1, and let G be a Hamiltonian graph of order n with n >= a + 2. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, a + 1]-factor if t(G) > a and has a Hamiltonian [a, b]-factor if t(G) >= a - 1 + a-1/b-2. This result is an improvement and extension of H. Enomoto and P. Katerinis's results (H. Enomoto, B. Jackson, P. Katerinis, A. Satio, Toughness and the existence of k-factors, Journal of Graph Theory 9(1985), 87-95; P. Katerinis, Toughness of graphs and the existence of factors, Discrete Mathematics 80(1990), 81-92).