Paths and Flows a Historical Survey

In 1927, the topologist Karl Menger published an article called Zur allgemeinen Kurventheorie (On the general theory of curves), in which he stated a remarkable result, now one of the most fundamental results in graph theory: Satz. Ist K ein kompakter regull ar eindimensionaler Raum, welcher zwi-schen den beiden endlichen Mengen P und Q n-punktig zusammenhh angend ist, dann enthh alt K n paarweise fremde BB ogen, von denen jeder einen Punkt von P und einen Punkt von Q verbindet. 1 P Q The result can be formulated as a maximum-minimum theorem in terms of graphs, as follows: Menger's theorem. Let G = (V VE) be an undirected 1 Theorem. If K is a compact regular one-dimensional space which is n-point connected between the two nite sets P and Q, then K contains n pairwise disjoint curves, each of which connec t s a p oint in P and a point in Q.