Some new sufficient conditions for graphs to have fractional k-factors

Let G be a graph, and k a positive integer. Let h : E(G) -> [0, 1] be a function. If Sigma(e(sic)x) h(e) = k holds for any x is an element of V (G), then we call G[Fh] a fractional k-factor of G with indicator function h, where F(h) = {e is an element of E(G) : h(e) > 0}. In this paper, we prove some results on the existence of a fractional k-factor in a simple graph depending on delta(G) and alpha(G). Furthermore, we show that the results are best possible in some sense.