The crossing number of complete bipartite graphs
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 $\displaystyle \left\lfloor\frac m2\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor \left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor.$
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— Daniel Brosch (@DanBrosch) Oct 8, 2026
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…
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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…