Improving the Crossing Lemma by finding more crossings in sparse graphs

Proceedings of the Annual Symposium on Computational Geometry

Volumn , Issue , 2004, Pages 68-75

  Pach, János ^a   Tardos, Gábor ^b   Radoičić, Radoš ^c   Tóth, Géza ^b  

^a NEW YORK UNIVERSITY   (United States)

^b ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS   (Hungary)

^c MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Theory

Indexed keywords

COMBINATORIAL MATHEMATICS; FUNCTIONS; GRAPH THEORY; MATHEMATICAL MODELS; PROBLEM SOLVING; THEOREM PROVING;

CROSSING LEMMA; JORDAN ARCS; MULTIGRAPHS; SPARSE GRAPHS;

COMPUTATIONAL GEOMETRY;

EID: 4544385055     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/997817.997831     Document Type: Conference Paper

References (19)

1. M. Ajtai, V. Chvátal, M. Newborn, E. Szemerédi, Crossing-free subgraphs, Ann. Discrete Mathematics 12 (1982), 9-12.
2. K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, E. Welzl, Combinatorial complexity bounds for arrangements of curves and surfaces, Discrete and Computational Geometry 5 (1990), 99-160.
3. E. Czabarka, O. Sýkora, L. Székely, I. Vrt'o, Biplanar crossing numbers I: A Survey of Results and Problems, manuscript.
4. T. K. Dey, Improved bounds for planar κ-sets and related problems, Discrete and Computational Geometry 19 (1998), 373-382.
5. G. Elekes, M. Nathanson, I. Z. Ruzsa, Convexity and sumsets, Journal of Number Theory 83 (2000). 194-201.
6. P. Erdös and R.K. Guy, Crossing number problems, American Mathematical Monthly 80 (1973), 52-58.
7. N. H. Katz and G. Tardos, Note on distinct sums and distinct distances, in: Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, American Mathematical Society, Providence, 2004
8. T. Leighton, Complexity Issues in VLSI, Foundations of Computing Series, MIT Press, Cambridge, MA, 1983.
9. J. Pach, Geometric graph theory, in: Surveys in Combinatorics, 1999 (J. D. Lamb and D. A. Preece, eds.), London Mathemetical Society Lecture Notes 267, Cambridge University Press, Cambridge, 1999, 167-200.
10. J. Pach, Geometric graph theory, Chapter 10 in: Handbook of Discrete and Computational Geometry, 2nd ed. (J. E. Goodman et al., edn.), CRC Press, Boca Raton, FL, 2004 (in press).
11. J. Pach and M. Sharir, On the number of incidences between points and curves, Combinatorics, Probability, and Computing, 7 (1998), 121-127.
12. J. Fach, J. Spencer, and G. Tóth, New bounds on crossing numbers, Discrete and Computational Geometry 24 (2000), 623-644.
13. J. Pach and G. Tardos, Isosceles triangles determined by a planar point set, Graphs and Combinatorics 18 (2002), 769-779.
14. J. Pach and G. Tóth, Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427-439.
15. J. Pach and G. Tóth, Which crossing number is it, anyway? Journal of Combinatorial Theory, Series B 80 (2000), 225-246.
16. J. Solymosi and Cs. Tóth, Distinct distances in the plane, Discrete Comput. Geom. 25 (2001), 629-634.
17. J. Solymosi, G. Tardos, and Cs. Tóth, The κ most frequent distances in the plane, Discrete and Computational Geometry 28 (2002), 769-779.
18. L. Székely, Crossing numbers and hard Erdös problems in discrete geometry, Combinatorics, Probability, and Computing, 6 (1997), 353-358.
19. E. Szemerédi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), 381-392.