Simplified existence theorems on all fractional [a, b]-factors

Let G be a graph with order n and let g, f : V (G) -> N such that g(v) <= f (v) for all v E, V (G). We say that G has all fractional (g.f)-factors if G has a fractional p-factor for every p : V(C) N such that g(v) <= p(v) < f (v) for every v E V(C). Let a < b be two positive integers. If g equivalent to a, f equivalent to b and G has all fractional (g,f)-factors, then we say that G has all fractional la, b I-factors. Suppose that n is sufficiently large for a and b. This paper contains two results on the existence of all (g,f)-factors of graphs. First, we derive from Anstee's fractional (g , j)-factor theorem a similar characterization for the property of all fractional (g, j)-factors. Second, we show that G has all fractional [a, b]-factors if the minimum degree is at least 1/4a ((a + b - 1)(2) + 4b) and every pair of nonadjacent vertices has cardinality of the neighborhood union at least bn/(a + b). These, lower bounds are sharp. (C) 2013 Elsevier B.V. All rights reserved.