Pictures of matchstick graphs

A matchstick graph is a planar graph with a plane embedding where all edges are of unit length. The name derives from the fact that they can be assembled on a flat surface out of matchsticks of equal length. Of particular interest are n-regular matchstick graphs. It’s easy to find 1-regular and 2-regular matchstick graphs (line segment and equilateral triangle, respectively). I’ll leave the case n = 3 as an exercise to the reader (it’s possible with 8 vertices and 12 edges, and not the first such graph you think of).

The smallest known 4-regular matchstick graph is the Harborth graph, with 52 vertices. It’s quite fun to make your own 4-regular matchstick graphs; mine ended up with 63 vertices:

4-regular graph

This has the additional property that the edges can be partitioned into 42 triangles, where each vertex is incident with two triangles. From this, it is therefore possible to derive a 3-regular planar graph of 42 vertices; this is obtained by doing the Δ-Y transform and contracting the resulting edges. Interestingly, this graph is itself a matchstick graph!

It has been proved that there are no finite 5-regular matchstick graphs, and the only 6-regular matchstick graph is the infinite triangular tiling (proof: consider Euler’s formula). On the surface of a sphere, however, the situation is different; there are a few 5-regular examples.

Snub polyhedra

The next object in this sequence is an infinite planar 5-regular matchstick graph.

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