Some sufficient conditions for graphs to have component factors

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