Systems of curves on surfaces
type: Item
instance of: scholarly article
title: Systems of curves on surfaces
zbMATH Open document ID: 0859.57014
author: Martin Juvan
author: Aleksander Malnič
author: Bojan Mohar
published in: Journal of Combinatorial Theory. Series B
publication date: 1996-12-03T00:00:00Z
review text: It is proved that for each compact (bordered) surface \(\Sigma\) and a positive integer \(k\) there is a constant \(N\) with the following property: If \(\Gamma\) is a family of pairwise nonhomotopic closed curves on \(\Sigma\) such that any two curves from \(\Gamma\) intersect in at most \(k\) points and each curve has at most \(k\) self-intersections, then \(\Gamma\) contains at most \(N\) curves. This result does not hold for families of curves with bounded algebraic intersection. Several examples are provided to show that the smallest possible \(N\) is rather large. The main result is used to show that for each compact surface \(\Sigma\) and each positive integer \(k\), there is only a finite number of distinct \(k\)-minimal triangulations of \(\Sigma\) and only a finite number of distinct embeddings of graphs in \(\Sigma\) that are minimal with respect to their face-width being equal to \(k\).
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 934821
zbMATH Keywords: homotopy
zbMATH Keywords: surface
zbMATH Keywords: closed curves
zbMATH Keywords: triangulations
zbMATH Keywords: embeddings of graphs
zbMATH Keywords: face-width
MaRDI profile type: Publication
OpenAlex ID: W2004380481
DOI: 10.1006/JCTB.1996.0053
description: scientific article; zbMATH DE number 934821