3-regular matchstick graphs with given girth
Sascha Kurz, Giuseppe Mazzuoccolo
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.