REMARKS ON COMPONENT FACTORS IN Κ1,r -FREE GRAPHS

An F-factor is a spanning subgraph Eta such that each connected component of Eta is isomorphic to some graph in F. We use P-k and Kappa(1,r) to denote the path of order k and the star of order r + 1, respectively. In particular, Eta is called a {P-2,P3}-factor of G if F = {P-2,P-3}; Eta is called a P >= k-factor of G if F = {P-k,Pk+1,...}, where k >= 2; Eta is called an S-n-factor of.. if F = {P-2,P-3, Kappa(1,3),...,..Kappa(1,n)}, where n >= 2. A graph.. is called a P >= k-factor covered graph if there is a P >= k-factor of G including e for any e is an element of E (G). We call a graph G is K-1,K-r -free if.. does not contain an induced subgraph isomorphic to K-1,K-r. In this paper, we give a minimum degree condition for the K-1,K-r-free graph with an S-n-factor and the. K-1,K-r-free graph with a P >= 3-factor, respectively. Further, we obtain sufficient conditions for K-1,K-r-free graphs to be P >= 2-factor P >==3-factor or {P-2,P-3}-factor covered graphs. In addition, examples show that our results are sharp.