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

Abstract

Let a,b,k be nonnegative integers with 2 ≤ a < b and 6 ≥ (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, b]-factor. In this paper, it is proved that a graph G is a fractional (a, b, k)-critical graph if G satisfies δ(G) ≥ a + k and S(G) ≥ 1(G) ≥ a - 1 + (a-1)(k+1)/b. Furthermore, it is showed that the result in this paper is best possible in some sense.