REMARKS ON COMPONENT FACTORS IN GRAPHS

For a family of connected graphs F, a spanning subgraph H of a graph G is called an F-factor of G if its each component is isomorphic to an element of F. In particular, H is called a S-k-factor of G if F = {K-1,K-1, K-1,K-2, ..., K-1,K-k}, where integer k >= 2; H is called a P->= 3-factor of G if every component in F is a path of order at least three. As an extension of S-k-factors, the induced star-factor (i.e., ISk-factor) is a spanning subgraph each component of which is an induced subgraph isomorphic to some graph in F = {K-1,K-1, K-1,K-2, ..., K-1,K-k}. In this paper, we firstly prove that a graph G has an S-k-factor if and only if its isolated toughness I(G) >= 1/k. Secondly, we prove that a planar graphs G has an S-2-factors if its minimum degree delta(G) >= 3. Thirdly, we give two sufficient conditions for graphs with ISk-factors by toughness and minimum degree, respectively. Additionally, we obtain three special classes of graphs admitting P->= 3-factors.