Wheel Graphs with Integer Edges

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Harborth's conjecture states that the edges of any planar graph can all have integer length. Planar graphs with non-triangular faces can have edges added to get a maximal planar (or triangulated) graph, where all faces are triangles. A solution for a given maximal planar graph would contain many integer wheel graphs. This Demonstration shows many integer wheel graphs found with searchs. Triangles are colored by their area radicals.

Contributed by: Ed Pegg Jr (March 2015)
Open content licensed under CC BY-NC-SA



Examining the hexagon possibilities took weeks, despite excluding many types of symmetries. A fast method to collect allowable heptagons, octagons, and general -gons with a given maximal edge length is currently unknown to the author.


[1] Wikipedia. "Harborth's Conjecture." (Mar 1, 2015) en.wikipedia.org/wiki/Harborth's_conjecture.

[2] Wikipedia. "Planar Graph." (Mar 1, 2015) en.wikipedia.org/wiki/Planar_graph# Maximal_planar _graphs.

[3] Wikipedia. "Robbins Pentagon." (Mar 1, 2015) en.wikipedia.org/wiki/Robbins_pentagon.

[4] Wikipedia. "Wheel Graph." (Mar 1, 2015) en.wikipedia.org/wiki/Wheel_graph.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.