Toughness and isolated toughness conditions for path-factor critical covered graphs

Given a graph G and an integer k = 2. A spanning subgraph H of G is called a P-=k-factor of G if every component of H is a path with at least k vertices. A graph G is said to be P-=k-factor covered if for any e ? E(G), G admits a P-=k-factor including e. A graph G is called a (P-=k, n)-factor critical covered graph if G - V' is P-=k-factor covered for any V' C V (G) with |V'| = n. In this paper, we study the toughness and isolated toughness conditions for (P-=k, n)-factor critical covered graphs, where k = 2, 3. Let G be a (n+ 1)-connected graph. It is shown that (i) G is a (P-=2, n)-factor critical covered graph if its toughness r (G) > n+2/3 ; (ii) G is a (P-=2, n)-factor critical covered graph if its isolated toughness I(G) > n+1/2 ; (iii) G is a (P-=3, n)-factor critical covered graph if r (G) = n+2/3 and |V(G)| = n + 3; (iv) G is a (P-=3, n)-factor critical covered graph if I(G) > n +32 and |V(G)| = n + 3. Furthermore, we claim that these conditions are best possible in some sense.