{"id":"openai-math/The-crossing-number-of-complete-bipartite-graphs","pdf":"https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026","tags":["graph-theory","crossing-number","combinatorics"],"score":0,"title":"The crossing number of complete bipartite graphs","domain":"Combinatorics","family":"165","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"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 $`K_{m,n}`$ is $`\\displaystyle \\left\\lfloor\\frac m2\\right\\rfloor \\left\\lfloor\\frac{m-1}{2}\\right\\rfloor \\left\\lfloor\\frac n2\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor.`$","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=determining%20the%20minimum%20number%20of%20crossings%20in%20plane%20drawings%20of%20complete%20and%20complete%20bipartite%20graphs","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"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…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://x.com/DanBrosch/status/2108279111814619281","date":"2026-10-08","kind":"x","added":"2026-10-24","author":"Daniel Brosch","handle":"DanBrosch"},{"url":"https://arxiv.org/abs/2610.25204","date":"2026-10-18","kind":"web","site":"arXiv.org","text":"Zarankiewicz's long standing conjecture states that the crossing number $cr(K_{m,n})$ of the complete bipartite graph is \\[ Z(m,n):=\\bigg\\lfloor{\\frac{m}{2}}\\bigg\\rfloor \\bigg\\lfloor{\\frac{m-1}{2}}\\bigg\\rfloor \\bigg\\lfloor{\\frac{n}{2}}\\bigg\\rfloor \\bigg\\lfloor{\\frac{n-1}{2}}\\bigg\\rfloor. \\] Using…","added":"2026-10-24","title":"Crossing numbers of complete bipartite graphs","author":"József Balogh, Nicholas Layman, Bernard Lidický, Florian Pfender et al.","record":"arxiv/2610.25204"}],"formalized":true,"family_title":"The Harary–Hill and Zarankiewicz crossing-number formulas"}