# Peter Dinkelacker's scientific contributions

## Publications (9)

Article
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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.
Article
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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.
Preprint
Full-text available
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.
Preprint
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.
Preprint
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.
Preprint
Full-text available
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.
Article
Full-text available
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 the authors present the latest known (4; n)-regular matchstick graphs for 4 ≤ n ≤ 11 with a minimum numb...
Article
Full-text available
A matchstick graph is a planar unit-distance graph. That 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 4-regular if every vertex has only degree 4. Examples of 4-regular matchstick graphs with less than 63 vertices are only known for 52, 54...
Article
Full-text available
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 ver...

## Citations

... For 52, 54, 57, 60, and 64 vertices only one example is known. For a proof we refer the reader to [5]. An overview of the currently known examples with 63 -70 vertices can be found in [3]. ...
... Proof. The results for d = 1 can be found in [11] and [12]. There are several possibilities to obtain 4-regular matchstick graphs of order n by combining examples of lower orders (see Figure 14). ...