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 ofℋ. 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 (1) a graph G has a { K 1,1 , K 1,2 , … , K 1, k }-factor if and only if bind( G ) ≥ 1/ k , where k ≥ 2 is an integer; (2) a connected graph G is a P ≥ 2 -factor covered graph if bind( G ) > 2/3; (3) 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.