Closing in on Hill's Conjecture
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 $0.996\, H(n)$.
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