Result 165, Combinatorics

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.

Lean formalization Proof

The bigger picture

Why it matters

How many edge crossings are unavoidable when a densely connected graph is drawn in the plane? Two unreviewed manuscripts claim exact answers for two basic graph families, showing that classical drawings cannot be improved by curving edges.

What changes?

Complete graphs join every pair of vertices; complete bipartite graphs join every vertex in one group to every vertex in the other. The reported minimum crossing counts are floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2)/4 for complete graphs on n vertices, and floor(m/2)*floor((m-1)/2)*floor(n/2)*floor((n-1)/2) for bipartite group sizes m and n. These claims cover all positive integer sizes and all plane drawings with continuous edge arcs. Here floor means rounding down, and * denotes multiplication.

What does that help mathematicians do?

The decisive consequence is a lower bound matching the classical constructions: no rearrangement of vertices or continuous rerouting of edges can beat their crossing counts. Researchers could also use these formulas as unavoidable-crossing certificates for larger graphs containing complete or complete bipartite subgraphs. Any drawing of the larger graph includes a drawing of that subgraph, so it inherits its minimum crossing requirement.

Are there practical applications?

The immediate value is foundational: the claimed formulas determine exactly how much crossing these two families force in the plane. They provide reference cases for studying graph drawings and distinguish unavoidable crossings from poor layout choices. The supplied material does not establish a layout algorithm for arbitrary graphs or a practical implementation benefit.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The crossing number of complete graphs

September 23, 2026 13 pages Main result formalized in Lean

We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is

14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋.\displaystyle \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-crossing-number-of-complete-graphs-September-23-2026,
  author = {{OpenAI}},
  title = {{The crossing number of complete graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf}{OAI:The-crossing-number-of-complete-graphs-September-23-2026}},
  year = {2026}
}

The crossing number of complete bipartite graphs

September 23, 2026 16 pages Main result formalized in Lean

We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph Km,nK_{m,n} is

⌊m2⌋⌊m−12⌋⌊n2⌋⌊n−12⌋.\displaystyle \left\lfloor\frac m2\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor \left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-crossing-number-of-complete-bipartite-graphs-September-23-2026,
  author = {{OpenAI}},
  title = {{The crossing number of complete bipartite graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf}{OAI:The-crossing-number-of-complete-bipartite-graphs-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/165.md.

The Harary–Hill and Zarankiewicz crossing-number formulas

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result proves Hill's proposed formula for the crossing number of the complete graph: for every n≥3n\ge3, cr(Kn)=14⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋\mathrm{cr}(K_n)=\frac14\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor\lfloor(n-2)/2\rfloor\lfloor(n-3)/2\rfloor. It proves the lower bound and constructs a matching two-page drawing. Crossings are spatial points, so repeated crossings of one edge pair count separately.

Zarankiewicz's crossing-number conjecture predicts cr(Km,n)=⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋\mathrm{cr}(K_{m,n})=\lfloor m/2\rfloor\lfloor(m-1)/2\rfloor\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor. The formalized result proves this equality for every positive m,nm,n and constructs a drawing attaining it. Crossings are counted as spatial points for continuous simple edge paths, including repeated crossings between the same pair of edges.

Comparator links

Result Comparator statement
Crossing number of complete graphs CompleteCrossing.lean
Crossing number of complete bipartite graphs BipartiteCrossing.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.