A Crossing Lemma for Multigraphs

Authors János Pach, Géza Tóth

Thumbnail PDF


  • Filesize: 0.5 MB
  • 13 pages

Document Identifiers

Author Details

János Pach
Géza Tóth

Cite AsGet BibTex

János Pach and Géza Tóth. A Crossing Lemma for Multigraphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Let G be a drawing of a graph with n vertices and e>4n edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in G is at least c{e^3 over n^2}, for a suitable constant c>0. In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound c{e^3 over mn^2}, where m denotes the maximum multiplicity of an edge in G. We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least c'{e^3 over n^2} for some c'>0, provided that the "lens" enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.
  • crossing number
  • Crossing Lemma
  • multigraph
  • separator theorem


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Miklós Ajtai, Vašek Chvátal, Monroe M Newborn, and Endre Szemerédi. Crossing-free subgraphs. North-Holland Mathematics Studies, 60(C):9-12, 1982. Google Scholar
  2. Noga Alon, Paul Seymour, and Robin Thomas. Planar separators. SIAM Journal on Discrete Mathematics, 7(2):184-193, 1994. Google Scholar
  3. Tamal K Dey. Improved bounds for planar k-sets and related problems. Discrete &Computational Geometry, 19(3):373-382, 1998. Google Scholar
  4. Michael R Garey and David S Johnson. Crossing number is np-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983. Google Scholar
  5. M. Kaufmann. Personal communication. Beyond-Planar Graphs: Algorithmics and Combinatorics, Schloss Dagstuhl, Germany, November 6-11, 2016. Google Scholar
  6. Frank Thomson Leighton. Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks. MIT press, Cambridge, 1983. Google Scholar
  7. J Pach, F Shahrokhi, and M Szegedy. Applications of the crossing number. Algorithmica, 16(1):111-117, 1996. Google Scholar
  8. János Pach and Géza Tóth. Thirteen problems on crossing numbers. Geombinatorics, 9:199-207, 2000. Google Scholar
  9. Marcus Schaefer. Complexity of some geometric and topological problems. In International Symposium on Graph Drawing, Lecture Notes in Computer Science, volume 5849, pages 334-344. Springer, 2010. Google Scholar
  10. Marcus Schaefer. The graph crossing number and its variants: A survey. The electronic journal of combinatorics, 1000:DS21, 2013. Google Scholar
  11. Micha Sharir and Pankaj K Agarwal. Davenport-Schinzel sequences and their geometric applications. Cambridge University Press, 1995. Google Scholar
  12. László A Székely. Crossing numbers and hard erdős problems in discrete geometry. Combinatorics, Probability and Computing, 6(3):353-358, 1997. Google Scholar
  13. László A Székely. A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Mathematics, 276(1-3):331-352, 2004. Google Scholar
  14. Endre Szemerédi and William T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381-392, 1983. Google Scholar