ISOLATED TOUGHNESS FOR PATH FACTORS IN NETWORKS

Let H be a set of connected graphs. Then an H-factor is a spanning subgraph of G, whose every connected component is isomorphic to a member of the set H. An H-factor is called a path factor if every member of the set H is a path. Let k >= 2 be an integer. By a P->= k-factor we mean a path factor in which each component path admits at least k vertices. A graph G is called a (P->= k, n)-factor-critical covered graph if for any W subset of V (G) with |W| = n and any e is an element of E(G - W), G - W has a P->= k-factor covering e. In this article, we verify that (i) an (n + lambda + 2)-connected graph G is a (P->= 2, n)-factor-critical covered graph if its isolated toughness I(G) > n+lambda+2/2 lambda+3 , where n and lambda are two nonnegative integers; (ii) an (n + lambda + 2)-connected graph G is a (P->= 3, n)-factor-critical covered graph if its isolated toughness I(G) > n+3 lambda+5 /2 lambda+3where n and lambda be two nonnegative integers.