On all fractional (a, b, k)-critical graphs

Let a, b, k, r be nonnegative integers with 1 a parts per thousand currency sign a a parts per thousand currency sign b and r a parts per thousand yen 2. Let G be a graph of order n with . In this paper, we first show a characterization for all fractional (a, b, k)-critical graphs. Then using the result, we prove that G is all fractional (a, b, k)-critical if and for any independent subset {x (1), x (2), aEuro broken vertical bar, x (r) } in G. Furthermore, it is shown that the lower bound on the condition is best possible in some sense, and it is an extension of Lu's previous result.