Global validity of Harborth’s conjecture for planar graphs

Establish whether Harborth’s conjecture holds for all planar graphs by proving that every planar graph admits an integral Fary embedding, i.e., a crossing-free straight-line drawing in which all edge lengths are integers.

Background

The paper studies truly integral Fary embeddings—planar straight-line drawings with integer edge lengths and integer grid vertex positions—and provides constructive, polynomial-area algorithms for trees and cactus graphs. These results contribute to the broader program surrounding Harborth’s conjecture (integral edge lengths) and Kleber’s conjecture (integral edge lengths with integer grid vertices).

Despite positive results for several subclasses (e.g., planar 3-regular graphs, max-degree-4 graphs that are non-4-regular, and planar 3-trees), the general case of Harborth’s conjecture remains a central question. The authors close with explicit open questions including whether Harborth’s conjecture is true for all planar graphs.