An isolated toughness condition for graphs to be fractional (a, b, k)-critical graphs

Let a, b,k be nonnegative integers with 2 <= a < b and b >= (a 1)(k +1). A graph G is called a fractional (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has a fractional [a, 4-factor. In this paper, it is proved that a graph G is a fractional (a, b, k)-critical graph if G satisfies delta(G) >= a + k and delta(G) >= I(G) >= a - 1 + (a-1)b(k+1)/b . Furthermore, it is showed that the result in this paper is best possible in some sense.