Combinatorial Applications

In this chapter, we use the theorems of Ford and Fulkerson about maximal flows to prove some central results in combinatorics. In particular, transversal theory can be developed from the theory of flows on networks; this approach was first suggested in the book by Ford and Fulkerson [FoFu62] and is also used in the survey [Jun86]. Compared with the usual approach [Mir71b] of taking Philip Hall’s marriage theorem [Hal35] – which we will treat in Section 7.3 – as the starting point of transversal theory, this way of proceeding has a distinct advantage: it also yields algorithms for explicit constructions.We shall study disjoint paths in graphs, matchings in bipartite graphs, transversals, the combinatorics of matrices, partitions of directed graphs, partially ordered sets, parallelisms, and the supply and demand theorem.