Two sufficient conditions for component factors in graphs

Let G be a graph. For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if each component of H is isomorphic to a member of ℋ. An ℋ-factor is also referred as a component factor. If G − e admits an ℋ-factor for any e∈ E(G), then we say that G is an ℋ-factor deleted graph. Let k ≥ 2 be an integer. In this article, we verify that (i) a graph G admits a {K1,1, K1,2, . . ., K1,k, (2k + 1)}-factor if and only if its binding number bindbind(G)≥22k+1\left( G \right) \ge {2 \over {2k + 1}}; (ii) a graph G with δ (G) ≥ 2 is a {K1,1, K1,2, . . ., K1,k, (2k + 1)}-factor deleted graph if its binding number bind(G) ≥ bind(G)≥22k-1\left( G \right) \ge {2 \over {2k - 1}}.