Subgraphs with prescribed valencies

Abstract

In this paper a generalization of the factor problem for finite undirected graphs is detailed. We prescribe certain inequalities for the valencies of a subgraph. We deduce formulas for the minimum “deviation” of this prescription and characterize the “optimally approaching” subgraphs. These results include the conditions of Tutte and Ore for the existence of a factor and the characterization of maximal independent edge-systems given in [3] and [11].