Disjoint Cycles in Directed Graphs on the Torus and the Klein Bottle

We give necessary and sufficient conditions for a directed graph embedded on the torus or the Klein bottle to contain pairwise disjoint circuits, each of a given orientation and homotopy, and in a given order. For the Klein bottle, the theorem is new. For the torus, the theorem was proved before by P. D. Seymour. This paper gives a shorter proof of that result.