Systems of Curves on Surfaces

Let 7 be a (bordered) compact surface and k an integer. Suppose that we have a set 1 of pairwise nonhomotopic simple (closed) curves in 7 with the property that any two curves from 1 intersect in at most k points. It is proved that 1 cannot contain too many curves; i.e., there is a number N depending only on 7 and k such that |1 | N (Theorems 3.3 and 3.4). The same holds for nonsimple curves as well (Theorem 3.5). This simple result does not seem to have a straightforward proof. It can be applied in the study of properties of graphs on surfaces. An example of such an application is presented in the last section. A special case when 7 is the torus is considered in Section 4 where we find linear upper bounds on the number of curves in 1. On the other hand, our bounds for surfaces of genus greater than 1 are probably far from being optimal. However, examples from Section 5 show that the bounds will not be very small, in general. In the last part of the paper we add an application of Theorem 3.3. We present a short proof of the fact that for each compact surface 7 and an integer k 0, there are only finitely many minor minimal embeddings in 7 of face-width k. This result has been verified previously (only for closed surfaces) with similar techniques but with longer proofs [MN, GRS]. Another application of Theorem 3.3 was obtained by Mohar and Robertson [MR] who considered the structure of nongenus embeddings of graphs in surfaces and proved that, up to certain generalized Whitney-type switchings, there are only a bounded number of types of nongenus embeddings in any fixed surface. article no. 0053