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

Abstract

Abstract 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 { | N G ( S ) | | S | : ∅ ≠ S ⊂ V ( G ) , N G ( 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.