Borrowing László Székely's lively expression, we show that Hill's conjecture is “asymptotically at least $98.5\%$ true.” This long-standing conjecture states that the crossing number cr$(K_n)$ of the complete graph $K_n$ is $H(n) := \frac{1}{4}{\lfloor\frac{n}{2}\rfloor}{\lfloor\frac{n-1}{2}\rfloor}{\lfloor\frac{n-2}{2}\rfloor}{\lfloor\frac{n-3}{2}\rfloor}$ for all $n\ge 3$. This has been verified only for $n\le 12$. Using the flag algebra framework, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large $n$, cr$(K_n) > 0.905\, H(n)$. Also using this framework, we prove that asymptotically cr$(K_n)$ is at least $0.985\, H(n)$. We also show that the spherical geodesic crossing number of $K_n$ is asymptotically at least $0.996\, H(n)$.

  • 1.  B. Ábrego O. Aichholzer S. Fernández-Merchant T. Hackl J. Pammer A. Pilz P. Ramos G. Salazar and  B. Vogtenhuber , All good drawings of small complete graphs, in Proceedings of the 31st European Workshop on Computational Geometry (EuroCG '15), Ljubljana , Slovenia , 2015 , pp. 57 -- 60 . Google Scholar

  • 2.  B. Ábrego O. Aichholzer S. Fernández-Merchant D. McQuillan B. Mohar P. Mutzel P. Ramos R. Richter and  B. Vogtenhuber , Bishellable drawings of $K_n$, in Proceedings of the XVII Encuentros de Geometría Computacional, Alicante , Spain , 2017 , pp. 17 -- 20 . Google Scholar

  • 3.  B. Ábrego O. Aichholzer S. Fernández-Merchant P. Ramos and  G. Salazar , More on the crossing number of $K_n$: Monotone drawings , Electron. Notes Discrete Math. , 44 ( 2013 ), pp. 411 -- 414 . CrossrefGoogle Scholar

  • 4.  B. M. Ábrego O. Aichholzer S. Fernández-Merchant P. Ramos and  G. Salazar , The $2$-page crossing number of $K_n$ , Discrete Comput. Geom. , 49 ( 2013 ), pp. 747 -- 777 . CrossrefISIGoogle Scholar

  • 5.  B. M. Ábrego O. Aichholzer S. Fernández-Merchant P. Ramos and  G. Salazar , Shellable drawings and the cylindrical crossing number of $K_n$ , Discrete Comput. Geom. , 52 ( 2014 ), pp. 743 -- 753 . CrossrefISIGoogle Scholar

  • 6.  B. M. Ábrego, O. Aichholzer, S. Fernández-Merchant, P. Ramos, and B. Vogtenhuber, Non-shellable drawings of $K_n$ with few crossings, in Proceedings of the 26th Annual Canadian Conference on Computational Geometry (CCCG '14), Halifax, Nova Scotia, Canada, 2014.Google Scholar

  • 7.  A. Arroyo D. McQuillan R. B. Richter and  G. Salazar , Levi's lemma, pseudolinear drawings of ${K}_n$, and empty triangles , J. Graph Theory , 87 ( 2018 ), pp. 443 -- 459 . CrossrefISIGoogle Scholar

  • 8.  A. Arroyo, D. McQuillan, R. B. Richter, and G. Salazar, Convex Drawings of the Complete Graph: Topology Meets Geometry, manuscript, 2017.Google Scholar

  • 9.  A. Arroyo D. McQuillan R. B. Richter and  G. Salazar , Drawings of $K_n$ with the same rotation scheme are the same up to triangle-flips (Gioan's theorem) , Australas. J. Combin. , 67 ( 2017 ), pp. 131 -- 144 . ISIGoogle Scholar

  • 10.  R. Baber and  J. Talbot , Hypergraphs do jump , Combin. Probab. Comput. , 20 ( 2011 ), pp. 161 -- 171 . CrossrefISIGoogle Scholar

  • 11.  M. Balko R. Fulek and  J. Kynčl , Crossing numbers and combinatorial characterization of monotone drawings of $K_n$ , Discrete Comput. Geom. , 53 ( 2015 ), pp. 107 -- 143 . CrossrefISIGoogle Scholar

  • 12.  J. Balogh P. Hu B. Lidický and  F. Pfender , Maximum density of induced $5$-cycle is achieved by an iterated blow-up of 5-cycle , European J. Combin. , 52 ( 2016 ), pp. 47 -- 58 , https://doi.org/10.1016/j.ejc.2015.08.006. CrossrefISIGoogle Scholar

  • 13.  J. Balogh P. Hu B. Lidický F. Pfender J. Volec and  M. Young , Rainbow triangles in three-colored graphs , J. Combin. Theory Ser. B , 126 ( 2017 ), pp. 83 -- 113 , https://doi.org/10.1016/j.jctb.2017.04.002. CrossrefISIGoogle Scholar

  • 14.  L. Beineke and  R. Wilson , The early history of the brick factory problem , Math. Intelligencer , 32 ( 2010 ), pp. 41 -- 48 . CrossrefISIGoogle Scholar

  • 15.  J. Blažek and  M. Koman , A minimal problem concerning complete plane graphs, in Theory of Graphs and Its Applications (Proc. Sympos. Smolenice, 1963), Publishing House of the Czechoslovak Academy of Sciences , Prague , 1964 , pp. 113 -- 117 . Google Scholar

  • 16.  R. Christian R. B. Richter and  G. Salazar , Zarankiewicz's conjecture is finite for each fixed m , J. Combin. Theory Ser. B , 103 ( 2013 ), pp. 237 -- 247 , https://doi.org/10.1016/j.jctb.2012.11.001. CrossrefISIGoogle Scholar

  • 17.  E. de Klerk J. Maharry D. V. Pasechnik R. B. Richter and  G. Salazar , Improved bounds for the crossing numbers of $K_{m,n}$ and $K_n$ , SIAM J. Discrete Math. , 20 ( 2006 ), pp. 189 -- 202 , https://doi.org/10.1137/S0895480104442741. LinkISIGoogle Scholar

  • 18.  E. de Klerk D. V. Pasechnik and  A. Schrijver , Reduction of symmetric semidefinite programs using the regular $\ast$-representation , Math. Program. , 109 ( 2007 ), pp. 613 -- 624 . CrossrefISIGoogle Scholar

  • 19.  E. Gethner L. Hogben B. Lidický F. Pfender A. Ruiz and  M. Young , On crossing numbers of complete tripartite and balanced complete multipartite graphs , J. Graph Theory , 84 ( 2017 ), pp. 552 -- 565 , https://doi.org/10.1002/jgt.22041. CrossrefISIGoogle Scholar

  • 20.  E. Gioan , Complete graph drawings up to triangle mutations, in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 3787, Springer , Berlin , 2005 , pp. 139 -- 150 , https://doi.org/10.1007/11604686_13. Google Scholar

  • 21.  X. Goaoc, A. Hubard, R. De Joannis De Verclos, J.-S. Sereni, and J. Volec, Limits of order types, in 31st International Symposium on Computational Geometry, LIPIcs. Leibniz Int. Proc. Inform. 34, Lars Arge and János Pach, eds., Schloss Dagstuhl Leibniz-Zentrum fur Informatik, Wadern, Germany, pp. 300--314.Google Scholar

  • 22.  R. K. Guy , The decline and fall of Zarankiewicz's theorem, in Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968), Academic Press , New York , 1969 , pp. 63 -- 69 . Google Scholar

  • 23.  R. K. Guy , Latest results on crossing numbers, in Recent Trends in Graph Theory (Proc. Conf., New York, 1970), Lecture Notes in Math. 186, Springer , Berlin , 1971 , pp. 143 -- 156 . Google Scholar

  • 24.  F. Harary and  A. Hill , On the number of crossings in a complete graph , Proc. Edinburgh Math. Soc. (2) , 13 ( 1962/1963 ), pp. 333 -- 338 , https://doi.org/10.1017/S0013091500025645. CrossrefGoogle Scholar

  • 25.  H. Hatami J. Hladký D. Král' S. Norin and  A. Razborov , On the number of pentagons in triangle-free graphs , J. Combin. Theory Ser. A , 120 ( 2013 ), pp. 722 -- 732 . CrossrefISIGoogle Scholar

  • 26.  D. J. Kleitman , The crossing number of $K_{5,n}$ , J. Combinatorial Theory , 9 ( 1970 ), pp. 315 -- 323 . CrossrefGoogle Scholar

  • 27.  D. Král' and  L. Mach . Sereni, A new lower bound based on Gromov's method of selecting heavily covered points , Discrete Comput. Geom. , 48 ( 2012 ), pp. 487 -- 498 . CrossrefISIGoogle Scholar

  • 28.  J. Kynčl , Improved enumeration of simple topological graphs , Discrete Comput. Geom. , 50 ( 2013 ), pp. 727 -- 770 . CrossrefISIGoogle Scholar

  • 29.  J. Kynčl , Simple realizability of complete abstract topological graphs simplified, in Graph Drawing and Network Visualization, Lecture Notes in Comput. Sci. 9411, Springer , Cham , 2015 , pp. 309 -- 320 , https://doi.org/10.1007/978-3-319-27261-0_26. Google Scholar

  • 30.  D. McQuillan S. Pan and  R. B. Richter , On the crossing number of $K_{13}$ , J. Combin. Theory Ser. B , 115 ( 2015 ), pp. 224 -- 235 . CrossrefISIGoogle Scholar

  • 31.  D. McQuillan and  R. B. Richter , A parity theorem for drawings of complete and complete bipartite graphs , Amer. Math. Monthly , 117 ( 2010 ), pp. 267 -- 273 . CrossrefISIGoogle Scholar

  • 32.  D. McQuillan and  R. B. Richter , On the crossing number of $K_n$ without computer assistance , J. Graph Theory , 82 ( 2016 ), pp. 387 -- 432 . CrossrefISIGoogle Scholar

  • 33.  J. W. Moon , On the distribution of crossings in random complete graphs , J. Soc. Indust. Appl. Math. , 13 ( 1965 ), pp. 506 -- 510 , https://doi.org/10.1137/0113032. LinkISIGoogle Scholar

  • 34.  S. Norin and Y. Zwols, Presentation at the BIRS Workshop on Geometric and Topological Graph Theory (13w5091), https://www.birs.ca/events/2013/5-day-workshops/13w5091/videos/watch/201310011538-Norin.html, 2013 (accessed: 2017-08-28).Google Scholar

  • 35.  J. Pammer, Rotation Systems and Good Drawings, Master's thesis, Graz University of Technology, Graz, Austria, 2014.Google Scholar

  • 36.  S. Pan and  R. B. Richter , The crossing number of $K_{11}$ is 100 , J. Graph Theory , 56 ( 2007 ), pp. 128 -- 134 . CrossrefISIGoogle Scholar

  • 37.  O. Pikhurko and  E. R. Vaughan , Minimum number of $k$-cliques in graphs with bounded independence number , Combin. Probab. Comput. , 22 ( 2013 ), pp. 910 -- 934 . CrossrefISIGoogle Scholar

  • 38.  A. A. Razborov and  Flag , J. Symbolic Logic , 72 ( 2007 ), pp. 1239 -- 1282 , https://doi.org/10.2178/jsl/1203350785. CrossrefISIGoogle Scholar

  • 39.  A. A. Razborov , On the minimal density of triangles in graphs , Combin. Probab. Comput. , 17 ( 2008 ), pp. 603 -- 618 , https://doi.org/10.1017/S0963548308009085. CrossrefISIGoogle Scholar

  • 40.  A. A. Razborov and  Flag : An interim report, in The Mathematics of Paul Erdös. II, Springer , New York , 2013 , pp. 207 -- 232 . Google Scholar

  • 41.  R. B. Richter and  C. Thomassen , Relations between crossing numbers of complete and complete bipartite graphs , Amer. Math. Monthly , 104 ( 1997 ), pp. 131 -- 137 . CrossrefISIGoogle Scholar

  • 42.  L. A. Székely brick factory problem: The status of the conjectures of Zarankiewicz and Hill, in Graph Theory---Favorite Conjectures and Open Problems. 1, Probl. Books in Math., Springer , Cham , 2016 , pp. 211 -- 230 . Google Scholar

  • 43.  P. Turán , A note of welcome , J. Graph Theory , 1 ( 1977 ), pp. 7 -- 9 . CrossrefGoogle Scholar

  • 44.  U. Wagner, On a geometric generalization of the upper bound theorem, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS '06), Berkeley, CA, 2006, pp. 635--645.Google Scholar