The MatchStickGraph is a type of graph that consists of a single edge and an equilateral triangle as its minimal 1- and 2-regular forms, respectively. The minimal known 4-regular MatchStickGraph is called the Harborth Graph, but it is yet to be proven optimal. Mathematica implements MatchStickGraphs as GraphData[%22Matchstick%22, n].
The minimal 1-regular matchstick graph is a single graph edge, and the minimal 2-regular graph is an equilateral triangle. The minimal known 4-regular matchstick graph is known as the Harborth graph (Hartsfield and Ringel 1994; Timm) but has not been proven optimal. These are illustrated above and summarized in the table below. The matchstick graphs are implemented in Mathematica as GraphData["Matchstick", n].
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