Equivalence of Harborth’s and Kleber’s conjectures and the effect of polynomial-area constraints

Determine whether Harborth’s conjecture (existence of integral Fary embeddings) and Kleber’s conjecture (existence of integral Fary embeddings with integer grid vertex coordinates) are equivalent for planar graphs, and ascertain whether the equivalence persists when restricting to truly integral Fary embeddings that occupy polynomial area.

Background

Harborth’s conjecture addresses the existence of planar straight-line drawings with integer edge lengths, whereas Kleber’s conjecture strengthens this by also requiring vertices to lie on integer grid points. The authors show constructive algorithms with area bounds for specific classes (trees and cacti) and highlight the broader relationship between these conjectures.

They explicitly ask whether the two conjectures are equivalent and how imposing polynomial-area constraints might affect this equivalence, indicating an unresolved foundational relationship in integral graph drawing.