It has 104 edges and 52 vertices and is currently the smallest known example of a 4-regular matchstick graph.

[8] The smallest 3-regular matchstick graph without triangles (girth ≥ 4) has 20 vertices, as proved by Kurz and Mazzuoccolo.

[9] The smallest known example of a 3-regular matchstick graph of girth 5 has 54 vertices and was first presented by Mike Winkler in 2019.

[12][13] More precisely, this problem is complete for the existential theory of the reals.

[15] It is also NP-complete to determine whether a matchstick graph has a Hamiltonian cycle, even when the vertices of the graph all have integer coordinates that are given as part of the input to the problem.

Uniformity of edge lengths has long been seen as a desirable quality in graph drawing,[19] and some specific classes of planar graphs can always be drawn with completely uniform edges.

Every tree can be drawn in such a way that, if the leaf edges of the tree were replaced by infinite rays, the drawing would partition the plane into convex polygonal regions, without any crossings.

In particular, it is possible to choose all edges to have equal length, resulting in a realization of an arbitrary tree as a matchstick graph.

[20] A similar property is true for squaregraphs, the planar graphs that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex either lies on the unbounded face or has at least four neighbors.

These graphs can be drawn with all faces parallelograms, in such a way that if a subset of edges that are all parallel to each other are lengthened or shortened simultaneously so that they continue to all have the same length, then no crossing can be introduced.

This makes it possible to normalize the edges so that they all have the same length, and obtain a realization of any squaregraph as a matchstick graph.