Abstract: Let h:E(G) →[0, 1] be a function. If a Σ _e∋x h(e) ≤ b holds for each x ∈ V(G), then we call G[F_h] a fractional[a, b]-factor of G with indicator function h, where F_h={e:e ∈ E(G), h(e) > 0}. A graph G is called a fractional[a, b]-covered graph if for every edge e of G, there is a fractional[a, b]-factor G[F_h] with h(e)=1. Zhou, Xu and Sun[S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (a, b, k)- critical covered graphs, Information Processing Letters 152(2019)105838] defined the concept of a fractional (a, b, k)-critical covered graph, i.e., for every vertex subset Q with|Q|=k of G, G-Q is a fractional[a, b]-covered graph. In this article, we study the problem of a fractional (2, b, k)-critical covered graph, and verify that a graph G with δ(G) ≥ 3 + k is a fractional (2, b, k)-critical covered graph if its toughness t(G) ≥ 1 + 1/b + k/2b, where b and k are two nonnegative integers with b ≥ 2 + k/2.

Key words: Graph, Toughness, Fractional[a, b]-factor, Fractional[a, b]-covered graph, Fractional (a, b, k)-critical covered graph