On the existence of subgraphs with degree constraints

Abstract

Let G be a graph on X, and let f(x), g(x) be positive integers; several authors have given conditions for the existence of a graph H on X, obtained from G by removing or duplicating edges, with degrees d H ( x) in the intervals [ g(x), f(x)] (for short, we shall say that H is a ( g, f)-graph). Most of the known conditions are complicated, and our purpose is to show that extremely simple conditions can be stated for the following cases: H is a ( kg, kf)-graph obtained from G by duplicating edges (Th. 1); H is a ( kg, kf)-graph obtained from G by duplicating edges and by removing edges (Th. 2); H is a (2 g, 2 f)-graph obtained from G by duplicating and removing edges that contains an arbitrary edge ((Th. 4); H is a (1, f)-graph obtained from G by removing edges (Th. 5); H is a graph obtained from G by removing edges so that the maximum degree and the minimum degree have a ratio less than k (Th. 5); H is a (1, f)-graph obtained from G by removing edges and containing an arbitrary edge (Th. 7).