Complete-factors and (g,f)-factors

A factor F of a graph is called a complete-factor if each component of F is complete. Let G be a graph, F be a complete-factor of G with ω(F)⩾2 and f,g be two integer-valued functions defined on V(G) with f(x)⩾g(x) for all x∈V(G). It is proved that if ω(F)≡0 (mod 2), or f(V(G)) even and f(x)≡g(x) (mod 2) for all x∈V(G), and if G−V(C) has a (g,f)-factor for each component C of F, then G has a (g,f)-factor. We show that the results in this paper are best possible.