An existence theorem on Hamiltonian (g, f)-factors in networks

Let a, b, and r be nonnegative integers with max{3, r + 1} ≤ a b − r, let G be a graph of order n, and let g and f be two integer-valued functions defined on V(G) with max{3, r + 1} ≤ a ≤ g(x) f(x) − r ≤ b − r for any x ∈ V(G). In this article, it is proved that if n ≥ (a+b−3)(a+b−5)+1 / a−1+r and bind(G) ≥ (a+b−3)(n-1)/(a−1+r)n−(a+b−3) , then G admits a Hamiltonian (g, f)-factor.