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

Given a graph Gand an integer k≥ 2. A spanning subgraph Hof Gis called a P≥k-factor of Gif every component of His a path with at least kvertices. A graph Gis said to be P≥k-factor covered if for any e∈ E(G), Gadmits a P≥k-factor including e. A graph Gis called a (P≥k, n)-factor critical covered graph if G– V′ is P≥k-factor covered for any V′ ⊆ 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 Gbe a (n+ 1)-connected graph. It is shown that (i) Gis a (P≥2, n)-factor critical covered graph if its toughness $ \tau (G)>\frac{n+2}{3}$; (ii) Gis a (P≥2, n)-factor critical covered graph if its isolated toughness $ I(G)>\frac{n+1}{2}$; (iii) Gis a (P≥3, n)-factor critical covered graph if $ \tau (G)>\frac{n+2}{3}$and |V(G)| ≥ n+ 3; (iv) Gis a (P≥3, n)-factor critical covered graph if $ I(G)>\frac{n+3}{2}$and |V(G)| ≥ n+ 3. Furthermore, we claim that these conditions are best possible in some sense.