Let G be a graph of order n, and let k \u0002 2 and m \u0002 0 be two integers. Let h : E(G) \u0003 [0, 1] be a function. If \u0002e∋x h(e) = k holds for each x \u0004 V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e \u0004 E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0 for any e \u0004 E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if \u0002(G) \u0002 k + m + m k+1 , n \u0002 4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)} \u0002 n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.<\/p>\r\n","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 55, 2011"}