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The Harary-Hill conjecture on the crossing number of complete graphs

Claims $\mathrm{cr}(K_n)=\frac14\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor\lfloor\frac{n-2}2\rfloor\lfloor\frac{n-3}2\rfloor$ for every $n$ , by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing number or the odd crossing number, which are different parameters.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Harary-Hill conjecture on the crossing number of complete graphs · Harary-Hill conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Harary-Hill conjecture on the crossing number of complete graphs

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No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: $\mathrm{cr}(K_n)=Z(n)$ for all $n\ge3$ under the ordinary point-count convention, with unrestricted vertex positions and edge routes. formalization.yaml lists OAI.Paper170.complete_graph_crossing_number (comparator CompleteCrossing) as the main result. Its statement defines admissible drawings in $\mathbb R^2$ (injective vertices, injective continuous edge paths whose interiors avoid vertices, finitely many crossing points, each locally homeomorphic to two axes, no triple points), counts crossing points, takes the infimum over drawings, and asserts it equals the floor-product formula for $n\ge3$ . That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. Hebbar and Mangam (2018, Int. J. Eng. Technol.) published earlier claims of both formulas, which the manuscripts cite and treat as upper-bound constructions only.