3-regular matchstick graphs with given girth

TL;DR

This paper investigates the existence of 3-regular planar matchstick graphs with specified girth, establishing existence conditions for girth 4 and providing an example for girth 5, thus advancing understanding of geometric graph configurations.

Contribution

The paper proves existence conditions for 3-regular planar matchstick graphs with girth 4 and presents a new example for girth 5, filling gaps in geometric graph theory.

Findings

Existence of girth 4 graphs for even n ≥ 20

Existence of a girth 5 graph with 180 vertices

Complete characterization for girth 3 and partial for girth 4

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.