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

Let a, b, k, r be nonnegative integers with 1 . a . b and r . 2. Let G be a graph of order n with n>(a+b)(r(a+b).2)+aka. 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 .(G).(r.1)b2a+k and |NG(x1).NG(x2)...NG(xr)|.bn+aka+b for any independent subset {x 1, x 2, ., x r } in G. Furthermore, it is shown that the lower bound on the condition |NG(x1).NG(x2)...NG(xr)|.bn+aka+b is best possible in some sense, and it is an extension of Lu.s previous result.