--- res: bibo_abstract: - 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⟶∞ .@eng bibo_authorlist: - foaf_Person: foaf_givenName: János foaf_name: Pach, János foaf_surname: Pach foaf_workInfoHomepage: http://www.librecat.org/personId=E62E3130-B088-11EA-B919-BF823C25FEA4 - foaf_Person: foaf_givenName: Gábor foaf_name: Tardos, Gábor foaf_surname: Tardos - foaf_Person: foaf_givenName: Géza foaf_name: Tóth, Géza foaf_surname: Tóth bibo_doi: 10.1007/978-3-030-68766-3_28 bibo_volume: 12590 dct_date: 2020^xs_gYear dct_isPartOf: - http://id.crossref.org/issn/0302-9743 - http://id.crossref.org/issn/1611-3349 - http://id.crossref.org/issn/9783030687656 dct_language: eng dct_publisher: Springer Nature@ dct_title: Crossings between non-homotopic edges@ ...