On fractional (f,n)(f,n)-critical graphs

Let G be a graph of order p  , and let a,ba,b and n   be nonnegative integers with b⩾a⩾2b⩾a⩾2, and let f   be an integer-valued function defined on V(G)V(G) such that a⩽f(x)⩽ba⩽f(x)⩽b for all x∈V(G)x∈V(G). A fractional f-factor is a function h that assigns to each edge of a graph G   a number in [0,1][0,1], so that for each vertex x   we have dGh(x)=f(x), where dGh(x)=∑e∋xh(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)(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)bind(G) is defined as follows,bind(G)=min{|NG(X)||X|:∅≠X⊆V(G),NG(X)≠V(G)}. In this paper, it is proved that G   is a fractional (f,n)(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.