Neighborhood conditions for the existence of ( g, f )-factors

Abstract

We obtain a sufficient condition for the existence of a (g, f)-factor in terms of vertex-deleted subgraphs. The following theorem is proved: Let G be a graph, k an even integer, g, f: V(G)→\mathbb{Z} two functions such that g(x)≤f(x) for all x∈V(G), and {u0, u1, …, uk/2} the set of distinct vertices of G such that {u1, u2, …, uk/2}⊆NG(u0). If g(u0)≤k≤f(u0) and G-{ui} has a (g, f)-factor for all i=0, …, k/2, then G has a (g, f)-factor.