Some sufficient conditions for graphs to have component factors

Let G denote a graph and k≥2 be an integer. A 1,1,K1,2,…,K1,k,T(2k+1)\-factor of G is a spanning subgraph, whose every connected component is isomorphic to an element of 1,1,K1,2,…,K1,k,T(2k+1)\, where T(2k+1) is one special family of tree. In this paper, we put forward some sufficient conditions for the existence of 1,1,K1,2,…,K1,k,T(2k+1)\-factors in graphs. Furthermore, we construct some extremal graphs to show that the main results in this paper are best possible.