Graphs drawn with few crossings per edge

Abstract

We show that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv. Fork≤4, we establish a better bound, (k+3)(v−2), which is tight fork=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.

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References

  1. [1]

    M. Ajtai, V. Chvátal, M. Newborn, andE. Szemerédi: Crossing-free subgraphs,Ann. Discrete Mathematics,12 (1982), 9–12.

    Google Scholar 

  2. [2]

    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, andE. Welzl: Combinatorial complexity bounds for arrangements of curves and surfaces,Discrete and Computational Geometry,5 (1990), 99–160.

    Google Scholar 

  3. [3]

    G. H. Hardy andE. M. Wright:An Introduction to the Theory of Numbers, University Press, Oxford, 1954.

    Google Scholar 

  4. [4]

    T. Leighton:Complexity Issues in VLSI, Foundations, of Computing Series, MIT Press, Cambridge, MA, 1983.

    Google Scholar 

  5. [5]

    J. Pach andP. K. Agarwal:Combinatorial Geometry, John Wiley, New York, 1995.

    Google Scholar 

  6. [6]

    J. Pach andM. Sharir: On the number of incidences between points and curves,Combinatorics, Probability, and Computing, to appear.

  7. [7]

    J. Spencer: A midrange crossing constant manuscript, 1997.

  8. [8]

    L. Székely: Crossing numbers and hard Erdős problems in discrete geometry,Combinatorics, Probability, and Computing, to appear.

  9. [9]

    E. Szemerédi andW. T. Trotter: Extremal problems in discrete geometry,Combinatorica,3 (1983), 381–392.

    Google Scholar 

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Additional information

Supported by NSF grant CCR-94-24398 and PSC-CUNY Research Award 667339.

Supported by OTKA-T-020914, OTKA-F-22234 and the Margaret and Herman Sokol Postdoctoral Fellowship Award.

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Pach, J., Tóth, G. Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997). https://doi.org/10.1007/BF01215922

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Mathematics Subject Classification (1991)

  • 05C10