Proof of the Harary--Hill conjecture: an exposition

Source abstract

The Harary--Hill conjecture states that the crossing number of the complete graph $K_n$ is exactly $\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor$. We present a streamlined variant of the proof of this conjecture recently announced by OpenAI (2026). Although the original proof is relatively short, it is quite difficult to follow. We simplify several steps and provide a concise and accessible exposition of the proof. We also combine this new result with known reductions to determine the crossing numbers of several other graph families.