Intersection reverse sequences and geometric applications

Adam Marcus, G. Tardos

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

Original languageEnglish
Pages (from-to)675-691
Number of pages17
JournalJournal of Combinatorial Theory, Series A
Volume113
Issue number4
DOIs
Publication statusPublished - May 2006

Fingerprint

Topological Graph
Reverse
Geometric Graphs
Intersection
Imply
Cycle
Curve
Arrangement
Incidence
Circle
Graph in graph theory
Vertex of a graph
Estimate
Observation

Keywords

  • Cyclic order
  • Extremal combinatorics
  • Pseudocircles
  • Topological graphs

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Intersection reverse sequences and geometric applications. / Marcus, Adam; Tardos, G.

In: Journal of Combinatorial Theory, Series A, Vol. 113, No. 4, 05.2006, p. 675-691.

Research output: Contribution to journalArticle

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