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Let G be a connected graph of order n and let a,b be two integers such that 2 ≤ a ≤ b. Let g and f be two integer-valued functions defined on V(G) such that a ≤ g(x) ≤ f(x) ≤ b for every x 驴 V(G). A spanning subgraph F of G is called a (g,f + 1)-factor if g(x) ≤ d F (x) ≤ f(x) + 1 for every x 驴 V(F). For a subset X of V(G), let $N_{G}(X)=\bigcup \limits _{x\in X}N_{G}(x)$. The binding number of G is defined by $bind(G)=min\{\frac{\mid N_{G}(X) \mid}{\mid X\mid}\mid \emptyset\neq X\subset V(G), N_{G}(X)\neq V(G)\}$. In this paper, it is proved that if $bind(G) \frac{(a+b)(n-1)}{an}$, f(V(G)) is even and $n\geq \frac{(a+b)^{2}}{a}$, then G has a connected (g,f + 1)-factor.

Binding Number and Connected (g,f + 1)-Factors in Graphs