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 l ess than or equal to a < b. We show that G has an [a,b]-factor if delt a(G) greater than or equal to a, n greater than or equal to 2a + b + a 2 - a/b and max{d(G)(u), d(G)(v)} greater than or equal to an/a + b fo r any two nonadjacent vertices u and v in G. This result is best possi ble, and it is an extension of T. lida and T. Nishimura's results (T. lida and T. Nishimura, An Ore-type condition for the existence of k-fa ctors in graphs, Graphs and Combinat. 7 (1991), 353-361; T. Nishimura, A degree condition for the existence of Ic-factors, J. Graph Theory 1 6 (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 cur rent results and problems on factors of graphs, Proc. 3rd China-USA In ternational Conference on Graph Theory and Its Application, Beijing (1 993). about connected [a, b]-factors is incorrect. (C) 1998 John Wiley & Sons, Inc.