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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.