Abstract

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 $0.996\, H(n)$.

Keywords

  1. crossing number
  2. complete graph
  3. Hill's conjecture
  4. flag algebras

MSC codes

  1. 05C10
  2. 05C62
  3. 68R10

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1261 - 1276
ISSN (online): 1095-7146

History

Submitted: 27 November 2017
Accepted: 23 April 2019
Published online: 18 July 2019

Keywords

  1. crossing number
  2. complete graph
  3. Hill's conjecture
  4. flag algebras

MSC codes

  1. 05C10
  2. 05C62
  3. 68R10

Authors

Affiliations

Funding Information

Consejo Nacional de Ciencia y Tecnología https://doi.org/10.13039/501100003141 : 222667
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1500121, DMS-1600390
University of Illinois at Urbana-Champaign https://doi.org/10.13039/100005302

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