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.
3-regular matchstick graphs with given girth
Mazzuoccolo, Giuseppe
2009-01-01
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.
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