@inproceedings{9299,
  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⟶∞ .},
  author       = {Pach, János and Tardos, Gábor and Tóth, Géza},
  booktitle    = {28th International Symposium on Graph Drawing and Network Visualization},
  isbn         = {9783030687656},
  issn         = {1611-3349},
  location     = {Virtual, Online},
  pages        = {359--371},
  publisher    = {Springer Nature},
  title        = {{Crossings between non-homotopic edges}},
  doi          = {10.1007/978-3-030-68766-3_28},
  volume       = {12590},
  year         = {2020},
}