A degree condition for a graph to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1 ≤ a < b. We show that G has an a, b]-factor if δ(G) ≥ a, n ≥ 2a + b + and max {d_G(u), d_G(v) ≥ for any two nonadjacent vertices u and v in G. This result is best possible, and it is an extension of T. Iida and T. Nishimura's results (T. Iida and T. Nishimura, An Ore-type condition for the existence of k-factors in graphs, Graphs and Combinat. 7 (1991), 353–361; T. Nishimura, A degree condition for the existence of k-factors, J. Graph Theory 16 (1992), 141–151). about the existence of a k-factor. As an immediate consequence, it shows that a conjecture of M. Kano (M. Kano, Some current results and problems on factors of graphs, Proc. 3rd China–USA International Conference on Graph Theory and Its Application, Beijing (1993). about connected a, b]-factors is incorrect. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 1–6, 1998