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 subset of V(G) with vertical bar Q vertical bar = k and any e is an element of 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.