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combinatorics / Topological graph theory; crossing numbers

The Harary-Hill conjecture on the crossing number of complete graphs

Claims cr(Kn)=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋\mathrm{cr}(K_n)=\frac14\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor\lfloor\frac{n-2}2\rfloor\lfloor\frac{n-3}2\rfloor for every nn, by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing number or the odd crossing number, which are different parameters.

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combinatoricsSep 23, 2026Significance 55/100Registry: lean checked

The Harary-Hill conjecture on the crossing number of complete graphs

Prior state unknown→proved

Claims cr(Kn)=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋\mathrm{cr}(K_n)=\frac14\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor\lfloor\frac{n-2}2\rfloor\lfloor\frac{n-3}2\rfloor for every nn, by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing…

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Claims cr(Kn)=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋\mathrm{cr}(K_n)=\frac14\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor\lfloor\frac{n-2}2\rfloor\lfloor\frac{n-3}2\rfloor for every nn, by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing number or the odd crossing number, which are different parameters.

Claims cr(Kn)=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋\mathrm{cr}(K_n)=\frac14\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor\lfloor\frac{n-2}2\rfloor\lfloor\frac{n-3}2\rfloor for every nn, by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing number or the odd crossing number, which are different parameters.

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