Toughness, Isolated Toughness and Path Factors in Graphs

A graph G is called a (P≥ n,k) -factor-critical covered graph if for any Q⊆ V(G) with $|Q|=k$ and any e∈ E(G-Q) , $G-Q$ has a P≥ n -factor covering e . We demonstrate that (i) a $(k+1)$ -connected graph G with at least $k+3$ vertices is a (P≥ 3,k) -factor-critical covered graph if its toughness t(G)>(2+k)/3 ; (ii) a $(k+2)$ -connected graph G is a (P≥ 3,k) -factor-critical covered graph if its isolated toughness I(G)>(5+k)/3 . Furthermore, we show that the conditions on $t(G)$ and $I(G)$ are sharp.