Graphs of G girth g with planar embeddings

3-regular matchstick graphs with given girth

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.