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 dF(x) = k for each x ∈ V (G), where dF(x) denotes the degree of x in F. For S ⊆ V (G), NG(S) = ∪x∊S NG(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.