Campus Units

Mathematics

Document Type

Article

Publication Version

Published Version

Publication Date

2019

Journal or Book Title

SIAM Journal on Discrete Mathematics

Volume

33

Issue

3

First Page

1261

Last Page

1276

DOI

10.1137/17M1158859

Abstract

Borrowing Laszlo Szekely'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(Kn) of the complete graph Kn is H(n) := 1 4 \lfloor n 2 \rfloor \lfloor n 1 2 \rfloor \lfloor n 2 2 \rfloor \lfloor n 3 2 \rfloor for all n \geq 3. This has been verified only for n \leq 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(Kn) > 0.905H(n). Also using this framework, we prove that asymptotically cr(Kn) is at least 0.985H(n). We also show that the spherical geodesic crossing number of Kn is asymptotically at least 0.996H(n).

Comments

This article is published as Balogh, József, Bernard Lidický, and Gelasio Salazar. "Closing in on Hill's Conjecture." SIAM Journal on Discrete Mathematics 33, no. 3 (2019): 1261-1276. doi: 10.1137/17M1158859. Posted with permission.

Copyright Owner

Society for Industrial and Applied Mathematics

Language

en

File Format

application/pdf

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