# Complete-graph drawing crossing reference

Family 165: The Harary–Hill and Zarankiewicz crossing-number formulas. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Diagram and graph-drawing developers can compare complete and complete-bipartite layouts with the claimed exact crossing formulas and matching drawings.

## Applicability and commercial boundary

Unrestricted continuous simple edge paths and spatial crossing counts differ from straight-line, orthogonal, fixed-node or obstacle-constrained layout costs.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract the two-page construction and define a renderer-compatible restricted benchmark.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scopes were additionally inspected; explicit effectiveness passages were inspected for 076, 143 and 178. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The crossing number of complete graphs](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf).