Document

# A Crossing Lemma for Multigraphs

## File

LIPIcs.SoCG.2018.65.pdf
• Filesize: 0.5 MB
• 13 pages

## Cite As

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)
https://doi.org/10.4230/LIPIcs.SoCG.2018.65

## Abstract

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.
##### Keywords
• crossing number
• Crossing Lemma
• multigraph
• separator theorem

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

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