Systems of Curves on Surfaces

Juvan, Martin; Malnič, Aleksander; Mohar, Bojan

doi:10.1006/jctb.1996.0053

Cited by 43 publications

(46 citation statements)

References 13 publications

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“…Juvan et al [17,Section 6] also study this concept in the case of surfaces with boundary; their proof technique implies that the number of irreducible triangulations is finite for every surface (possibly with boundary). For a more detailed survey on results on irreducible triangulations, see Mohar and Thomassen [22,Sect.…”

Section: Previous Work For Surfaces Without Boundarymentioning

confidence: 99%

Irreducible Triangulations of Surfaces with Boundary

Boulch

Verdière

Nakamoto

2012

Graphs and Combinatorics

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A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O (g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.

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Section: Previous Work For Surfaces Without Boundarymentioning

confidence: 99%

Irreducible Triangulations of Surfaces with Boundary

Boulch

Verdière

Nakamoto

2012

Graphs and Combinatorics

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“…Systems of closed curves on surfaces have been studied in the past, see for instance [19]. For a related concept of a complex of curves see [17,22].…”

Section: Complete Curve Arrangements On Surfacesmentioning

confidence: 99%

Oriented matroids and complete-graph embeddings on surfaces

Bokowski

Pisanski

2007

Journal of Combinatorial Theory, Series A

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We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g + 1 curves in contrast to the non-orientable surface where the number of curves is not bounded.

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“…Questions about the combinatorial properties of collections of curves on surfaces with specified pairwise intersection data have been considered by various authors. In [JMM96], the first explicit bounds for the maximum size of any curve collection such that any pair has geometric intersection number at most k were obtained for surfaces of finite-type. In the remarkably delicate case k = 1, increasingly precise bounds on these maximum sizes are given in [MRT14], [Prz15], [ABG17], and [Gre18a], but aside from several low-complexity examples, exact values remain unknown.…”

Section: Introductionmentioning

confidence: 99%

Algebraic k-systems of curves

et al. 2020

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A collection ∆ of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number α, β is equal to k in absolute value for every α, β ∈ ∆ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g + 1 when g ≥ 3 or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as g 2 .

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Systems of Curves on Surfaces

Abstract: It is proved that for each compact (bordered) surface 7 and each integer k there is a constant N with the following property: If 1 is a family of pairwise nonhomotopic closed curves on 7 such that any two curves from 1 intersect in at most k points, then 1 contains at most N curves.

Cited by 43 publications

References 13 publications

Irreducible Triangulations of Surfaces with Boundary

Irreducible Triangulations of Surfaces with Boundary

Oriented matroids and complete-graph embeddings on surfaces

Algebraic k-systems of curves

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