In geometric graph theory, a branch of mathematics, a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other. That is, it is a graph that has an embedding which is simultaneously a unit distance graph and a plane graph. Informally, matchstick graphs can be made by placing noncrossing matchsticks on a flat surface, hence the name.
Regular matchstick graphs
It is known that there are matchstick graphs that are regular of any degree up to 4. The complete graphs with one, two, and three vertices (a single vertex, a single edge, and a triangle) are all matchstick graphs and are 0-, 1-, and 2-regular respectively. The smallest 3-regular matchstick graph is formed from two copies of the diamond graph placed in such a way that corresponding vertices are at unit distance from each other; its bipartite double cover is the 8-crossed prism graph.
In 1986, Heiko Harborth presented the graph that would bear his name, the Harborth Graph. With 104 edges and 52 vertices, is the smallest known example of a 4-regular matchstick graph. It is a rigid graph.
Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62. The graphs with 54, 57, 65, 67, 73, 74, 77 and 85 vertices were first published in 2016. For 52, 54, 57, 60 and 64 vertices only one example is known. Of these five graphs only the one with 60 vertices is flexible, the other four are rigid.
It is not possible for a regular matchstick graph to have degree greater than four.
The smallest 3-regular matchstick graph without triangles (girth ≥ 4) has 20 vertices, as proved by Kurz and Mazzuoccolo. Furthermore, they exhibit the smallest known example of a 3-regular matchstick graph of girth 5 (180 vertices).
It is NP-hard to test whether a given undirected planar graph can be realized as a matchstick graph. More precisely, this problem is complete for the existential theory of the reals. Kurz (2014) provides some easily tested necessary criteria for a graph to be a matchstick graph, but these are not also sufficient criteria: a graph may pass Kurz's tests and still not be a matchstick graph.
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.
The numbers of distinct (nonisomorphic) matchstick graphs are known for 1, 2, 3, ... up to ten edges; they are:
Special classes of graphs
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. For such a drawing, if the lengths of each edge are changed arbitrarily, without changing the slope of the edge, the drawing will remain planar. 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.
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. In particular it is possible to normalize the edges so that they all have the same length, and obtain a realization of any squaregraph as a matchstick graph.
Related classes of graphs
Every matchstick graph is a unit distance graph. Penny graphs are the graphs that can be represented by tangencies of non-overlapping unit circles. Every penny graph is a matchstick graph. However, some matchstick graphs (such as the eight-vertex cubic matchstick graph of the first illustration) are not penny graphs, because realizing them as a matchstick graph causes some non-adjacent vertices to be closer than unit distance to each other.
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