Multicommodity flows in graphs

Abstract Suppose that G is a graph, and ( s i , t i ) (1≤ i ≤ k ) are pairs of vertices; and that each edge has a integer-valued capacity (≥0), and that q i ≥0 (1≤ i ≤ k ) are integer-valued demands. When is there a flow for each i , between s i and t i and of value q i , such that the total flow through each edge does not exceed its capacity? Ford and Fulkerson solved this when k =1, and Hu when k =2. We solve it for general values of k , when G is planar and can be drawn so that s 1 ,…, s l , t 1 , …, t l ,…, t l are all on the boundary of a face and s l +1 , …, S k , t l +1 ,…, t k are all on the boundary of the infinite face or when t 1 =⋯= t l and G is planar and can be drawn so that s l +1 ,…, s k , t 1 ,…, t k are all on the boundary of the infinite face. This extends a theorem of Okamura and Seymour.