3-regular matchstick graphs with given girth

Abstract

We consider 3-regular planar matchstick graphs, i.e. those which have a planar embedding such that all edge lengths are equal, with given girth g. For girth 3 it is known that such graphs exist if and only if the number of vertices n is an even integer larger or equal to 8. Here we prove that such graphs exist for girth g=4 if and only if n is even and at least 20. We provide an example for girth g=5 consisting of 180 vertices.

18 pages, 1 table, 8 figures

References in corpus (1)

• Minimal Polynomials for the Coordinates of the Harborth Graph

Cited by in corpus (5)

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• Minimal completely asymmetric (4; n)-regular matchstick graphs
• A lower bound for 4-regular planar unit distance graphs