Abstract:

 Let G be a graph with 1-factor F and let g and f be two non-negative integer-valued functions defined on V(G) such that g(x)≤d_G(x)for every x∈V(G). Suppose f(V(G)) is even.

(1)    If f(x)=f(y) and G-{x,y} has a (g,f)-factor for every xy∈F, then G itself has a (g,f)-factor.

(2) If f(x)=f(y) and G-{x,y} has a f-factor for every xy∈F, then G itself has a ffactor.

Key words: Ｇｒａｐｈ, Ｓｕｂｇｒａｐｈ, Ｆａｃｔｏｒ