\(f\)-optimal factors of infinite graphs (Q1191924)

Given an infinite graph \(G=(V,E)\) and a function \(f\) from \(V\) into the set of natural numbers, a perfect \(f\)-factor is a subgraph \(H=(V,F)\) with \(F\subseteq E\) such that \(\deg_ H(v)=f(v)\) for each vertex \(v\in V\). A necessary and sufficient condition for the existence of perfect \(f\)- factors of countable graphs is proved. Also nonperfect factors which approximate \(f\) optimally are studied.