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. © 1993 by Academic Press, Inc

Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.Comment: 15 pages, 7 figure

Disjoint circuits on a Klein bottle and a theorem on posets

In this paper, we consider the problem of packing disjoint directed circuits in a digraph drawn on the Klein bottle or on the torus. We formulate a problem on posets which unifies all the problems considered by Ding et al. and by Seymour. Then we generalize all the results of their two papers by proving a theorem on our special posets. © 1993

Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.Comment: 76 page, 48 figures; referee comments incorporated and typos fixed; accepted at TAM