Binding Number and Connected ( g, f+1)-factors in graphs

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 is an element of 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 is an element of V (F). For a subset X of V (G), let N-G (X) boolean OR N-G(x). The binding number of G is defined by bind(G) = min{N-G(X)(x is an element of X)vertical bar/vertical bar X vertical bar vertical bar theta not equal X subset of V(G), N-G (X) not equal V(G)}. In this paper, it is proved that if bind(G) >(a+b)(n-1)/an, f (V(G)) is even and n >= then (a+b)(2)/a, has a connected (g, f + 1)-factor.