A remark about 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 each 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], so that for each vertex x we have dG(x) = f(x), where d h G(x) = ∑ e3x 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{ |NG(X)| |X| : ∅ 6= X ⊆ V (G), NG(X) 6= V (G)}. In this paper, it is proved that G is a fractional (f, n)-critical graph if p ≥ (a+b−1)(a+b−2)−2 a + bn a−1 , bind(G) ≥ (a+b−1)(p−1) a(p−1)−bn and δ(G) 6= b (b−1)p+a+b+bn−2 a+b−1 c.