Combinatorics · Oct 6, 2026

The Harary–Hill and Zarankiewicz crossing-number formulas

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.

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Unreleased OpenAI model

AI systems: Unreleased OpenAI model

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