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15

## Project log

In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is ≤ 6422.
A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. 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. In this article we present 38 different examples with 50 - 62 vertices which contain two, three, or four distances which differ slightly from the unit length. These graphs should show why this subject is so extraordinarily difficult to deal with and should also be an incentive for the interested reader to find solutions for the missing numbers of vertices.
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is $\leq$6422.
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is ≤ 6422.
The first part (pp. 1 - 7) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (pp. 8 - 15) presents the currently known examples of (2; 4)-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2.
Es wird die Konstruktion eines neuen 4-regulären Streichholzgraphen mit 114 Kanten vorgestellt, welcher vom Autor am 15. April 2016 entdeckt wurde.
A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph (m; n)-regular if every vertex has only degree m or n. In this article we present the latest known (4; n)-regular matchstick graphs for 4 ≤ n ≤ 11 with a minimum number of vertices and a completely asymmetric structure. We call a matchstick graph completely asymmetric, if the following conditions are complied. 1) The graph is rigid. 2) The graph has no point, rotational or mirror symmetry. 3) The graph has an asymmetric outer shape. 4) The graph can not be decomposed into rigid subgraphs and rearrange to a similar graph which contradicts to any of the other conditions.