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.
released 2026-09-23 | 1 theorem · 7 lemmas · 10 proofs · 6,180 words |
PLAY LEVEL 1 »(pdf)
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is
$\displaystyle \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor.$
released 2026-09-23 | 1 theorem · 6 lemmas · 7 proofs · 6,595 words |
PLAY LEVEL 2 »(pdf)
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
$\displaystyle \left\lfloor\frac m2\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor \left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor.$