Harborth's conjecture states that every planar graph has a planar drawing in which all edge lengths are integers. I was looking at that, and I wondered what was known about hexagonal grid graphs. For example, the following graph might be tricky.
Each of those hexagons would be a hexagonal grid graph. My question is how much variety is possible for an infinite hexagonal grid? I can only think of two classes:
- Identical integer triangles, all neighboring triangles with edge rotation.
- Rotate, Rotate, Mirror for identical triangles. Here's the 3-4-5 triangle mirrored on 5.
Are there other ways to get a hexagonal grid where all edge lengths are integers?
EDIT: I've started building a database of integer wheel graphs.
