--- _id: '9299' abstract: - lang: eng text: We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ . acknowledgement: Supported by the National Research, Development and Innovation Office, NKFIH, KKP-133864, K-131529, K-116769, K-132696, by the Higher Educational Institutional Excellence Program 2019 NKFIH-1158-6/2019, the Austrian Science Fund (FWF), grant Z 342-N31, by the Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926, and by the ERC Synergy Grant “Dynasnet” No. 810115. A full version can be found at https://arxiv.org/abs/2006.14908. article_processing_charge: No author: - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach - first_name: Gábor full_name: Tardos, Gábor last_name: Tardos - first_name: Géza full_name: Tóth, Géza last_name: Tóth citation: ama: 'Pach J, Tardos G, Tóth G. Crossings between non-homotopic edges. In: 28th International Symposium on Graph Drawing and Network Visualization. Vol 12590. LNCS. Springer Nature; 2020:359-371. doi:10.1007/978-3-030-68766-3_28' apa: 'Pach, J., Tardos, G., & Tóth, G. (2020). Crossings between non-homotopic edges. In 28th International Symposium on Graph Drawing and Network Visualization (Vol. 12590, pp. 359–371). Virtual, Online: Springer Nature. https://doi.org/10.1007/978-3-030-68766-3_28' chicago: Pach, János, Gábor Tardos, and Géza Tóth. “Crossings between Non-Homotopic Edges.” In 28th International Symposium on Graph Drawing and Network Visualization, 12590:359–71. LNCS. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-68766-3_28. ieee: J. Pach, G. Tardos, and G. Tóth, “Crossings between non-homotopic edges,” in 28th International Symposium on Graph Drawing and Network Visualization, Virtual, Online, 2020, vol. 12590, pp. 359–371. ista: 'Pach J, Tardos G, Tóth G. 2020. Crossings between non-homotopic edges. 28th International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network VisualizationLNCS vol. 12590, 359–371.' mla: Pach, János, et al. “Crossings between Non-Homotopic Edges.” 28th International Symposium on Graph Drawing and Network Visualization, vol. 12590, Springer Nature, 2020, pp. 359–71, doi:10.1007/978-3-030-68766-3_28. short: J. Pach, G. Tardos, G. Tóth, in:, 28th International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2020, pp. 359–371. conference: end_date: 2020-09-18 location: Virtual, Online name: 'GD: Graph Drawing and Network Visualization' start_date: 2020-09-16 date_created: 2021-03-28T22:01:44Z date_published: 2020-09-20T00:00:00Z date_updated: 2021-04-06T11:32:32Z day: '20' department: - _id: HeEd doi: 10.1007/978-3-030-68766-3_28 external_id: arxiv: - '2006.14908' intvolume: ' 12590' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2006.14908 month: '09' oa: 1 oa_version: Preprint page: 359-371 project: - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize publication: 28th International Symposium on Graph Drawing and Network Visualization publication_identifier: eissn: - 1611-3349 isbn: - '9783030687656' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' series_title: LNCS status: public title: Crossings between non-homotopic edges type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 12590 year: '2020' ...