The crossing number of complete graphs
We prove the Harary–Hill conjecture: for every positive integer n , the ordinary crossing number of the complete graph K n is $\displaystyle \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor.$
Comments and reviews
No comments yet.
Discussed elsewhere
— 早稲田大学 早水桃子研究室 (@hayamizu_lab) Oct 8, 2026
API
curl /api/v1/records/openai-math/The-crossing-number-of-complete-graphs
curl -X POST /api/v1/records/openai-math/The-crossing-number-of-complete-graphs/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/openai-math/The-crossing-number-of-complete-graphs/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'
Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
determining the minimum number of crossings in plane drawings of complete and complete bipartite graphs. Barnette’s Hamiltonian-cycle conjecture. In Coloring Simple Polytopes and Triangulations I discussed Barnette’s conjecture that every cubic, bipartite, 3-connected planar graph has a…