Harborth's conjecture

[1][2][3] This conjecture is named after Heiko Harborth, and (if true) would strengthen Fáry's theorem on the existence of straight-line drawings for every planar graph.

For this reason, a drawing with integer edge lengths is also known as an integral Fáry embedding.

However, for the outerplanar graphs a more direct construction of integral Fáry embeddings is possible, based on the existence of infinite subsets of the unit circle in which all distances are rational.

If such a subset existed, it would form a universal point set that could be used to draw all planar graphs with rational edge lengths (and therefore, after scaling them appropriately, with integer edge lengths).

[14] According to the Erdős–Anning theorem, infinite non-collinear point sets with all distances being integers cannot exist.