On fractional (f, n)-critical graphs

Let G be a graph of order p, and let a, b and n be nonnegative integers with b >= a >= 2. and let f be an integer-valued function defined on V(G) such that a <= f(x) <= b for all x is an element of V (G). A fractional f-factor is a function h that assigns to each edge of a graph G a number in |0, 1|, so that for each vertex x we have d(G)(h)(x) = f(x), where d(G)(h)(x) = Sigma(e(sic)x) h(e) (the sum is taken over all edges incident to x) is a fractional degree of x in G. Then a graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. The binding number bind(G) is defined as follows, bind(G) = min {|N-G(X)|/|x| : 0 not equal X subset of V(G), N-G(X) not equal V(G)}. In this paper, it is proved that G is a fractional (f, n)-critical graph if bind(G) > (a + b - 1)(p - 1)/(ap - (a + b) - bn + 2) and p >= (a + b)(a + b - 3)/a + bn/(a - 1). Furthermore. it is showed that the result in this paper is best possible in some sense. (C) 2009 Elsevier B.V. All rights reserved.