The crossing number of the complete graph K13 equals the Harary-Hill formula value 225.

Statement

$\mathrm{cr}(K_{13}) = 225$, the Harary-Hill value $\tfrac14 \lfloor \tfrac{n}{2} \rfloor \lfloor \tfrac{n-1}{2} \rfloor \lfloor \tfrac{n-2}{2} \rfloor \lfloor \tfrac{n-3}{2} \rfloor$ at $n = 13$; the conjecture is proved only through $n = 12$.

Assessment

Renown 3/5

The concrete frontier of a 65-year-old drawing conjecture.

Attackability 2/5

A refutation is a single drawing of K13 with fewer than 225 crossings -- a finite, instantly checkable witness; decades of SAT-assisted and heuristic drawing searches have matched, never beaten, the bound.

finite witness

5/5

oracle cost

5/5

freshness

1/5

seedability

3/5

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