A sufficient condition for the existence of a k-factor excluding a given r-factor

Let G be a graph, and let k, r be nonnegative integers with k ≥ 2. A k-factor of G is a spanning subgraph F of G such that d_F(x) = k for each x ∈ V (G), where d_F(x) denotes the degree of x in F. For S ⊆ V (G), N_G(S) = ∪_x∊SN_G(x). The binding number of G is defined by bind(G)=min{|NG(S)||S|:∅≠S⊂V(G),NG(S)≠V(G)}$\begin{array}{} (G) = {\rm{min }}\{ \frac{{|{N_G}(S)|}}{{|S|}}:\emptyset \ne S \subset V(G),{N_G}(S) \ne V(G)\} \end{array}$. In this paper, we obtain a binding number and neighborhood condition for a graph to have a k-factor excluding a given r-factor. This result is an extension of the previous results.