Closing in on Hill's conjecture
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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).
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.