SOME RESULTS ABOUT COMPONENT FACTORS IN GRAPHS

For a set H of connected graphs, a spanning subgraph H of a graph G is called an H-factor of G if every component of H is isomorphic to a member of H. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P->= k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k - 1). A graph G is called a P->= k-factor covered graph, if for each edge e of G, there is a P->= k-factor covering e. In this paper, we prove that (i) a graph G has a K-1,(1), K-1,K-2, ..., K-1,K-k}-factor if and only if bind(G) >= 1/k, where k >= 2 is an integer; (ii) a connected graph G is a P->= 2-factor covered graph if bind(G) > 2/3; (iii) a connected graph G is a P->= 3-factor covered graph if bind(G) >= 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.