# Wheel Graphs with Integer Edges

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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 searches. Triangles are colored by their area radicals.

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Contributed by: Ed Pegg Jr (March 2015)

Open content licensed under CC BY-NC-SA

## Details

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.

References

[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.

[5] Gábor Damásdi, "Odd Wheels Are Not Odd-Distance Graphs." Disc. & Comp. Geom. vol 69, pp327–337, 2023. https://link.springer.com/article/10.1007/s00454-021-00325-0.

## Snapshots

## Permanent Citation