Year: 2019
Type: article
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 $0.996\, H(n)$.
DOI: 10.1137/17m1158859
Authors: József Balogh, Bernard Lidický, Gelasio Salazar
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2770549264