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Approximate Solutions of 4-regular Matchstick Graphs with 50 - 62 Vertices

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Abstract and Figures

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.
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arXiv:1906.11908v3 [math.GM] 13 Oct 2020
Approximate Solutions of 4-regular Matchstick Graphs
with 50 – 62 Vertices
A catalog of examples which contain 2 or 3 forbidden distances
Mike Winkler
Fakult¨at f¨ur Mathematik
Ruhr-Universit¨at Bochum, Germany
mike.winkler@ruhr-uni-bochum.de
www.mikematics.de
June 14, 2020
Abstract
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.
Figure 1: 51 vertices
1
1 Introduction
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, whereby 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 are currently known for all number of vertices 52
except for 53, 55, 56, 58, 59, 61, and 62. 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]. The currently smallest known example with 52 vertices is the
so-called Harborth Graph presented first by Heiko Harborth in 1986 [1].
Figure 2: The Harborth Graph
In this article we present 38 different examples of 4-regular rigid planar graphs for all number of
vertices 50 and 62. These graphs are not matchstick graphs, because each graph contains at least
two edges which differ slightly from the unit length. The edges which do not have unit length are
colored red.
The geometry of the graphs has been verified with the software MATC HSTICK GRA PH S CAL CU -
LATOR (MGC) [2]. This remarkable software created by Stefan Vogel runs directly in web browsers1.
A special version of the MGC contains all graphs from this article and is available on the author’s
website2. The graphs were constructed and first presented by the author between March 21, 2014 and
June 5, 2020 in a graph theory internet forum [6] [7].3
1For optimal functionality and design please use the Firefox web browser.
2http://mikematics.de/matchstick-graphs-calculator.htm
3Except Figure 27 by Peter Dinkelacker.
2
2 Construction rules and Epsilon graphs
To get a kind of fair approximate solutions we constructed the graphs by using the following rules.
The graph must be rigid.
Equilateral triangles which contain vertices of the outer circle of the graph may not contain
forbidden distances. These set of equilateral triangles we denote as the frame of the graph.
The graph may not contain more than three forbidden distances.
Whenever possible the forbidden distances may not deviate more than 10 percent from the unit
length.
The exceptions from the last construction rule apply to the number of vertices for which only one
approximate example currently have been found or graphs that we do not want to deprive the interested
reader.
Please note that much better approximate solutions are possible if we would ignore the second
and/or the third construction rule. The deviation from the unit length of an edge with a forbidden
distance becomes smaller the closer these edge is to the outer circle of the graph, or if we distributing
the deviation on all edges of the graph.
Without the first three construction rules it is possible to construct flexible graphs whose forbidden
distances can be infinitesimally approximated to the unit length. These kind of graphs we will denote
as Epsilon graphs. Figure 2 shows the smallest possible example with a minimum number of vertices.
This Epsilon graph has 27 vertices, a rotational symmetry of order 3, and contains six forbidden
distances of equal length.
red edges 0.845 red edges = 1 + ε
Figure 3: 27 vertices
3
Figure 3 and 4 show the smallest known example with only 4 forbidden distances. These Epsilon
graph has 42 vertices and a point symmetry.
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Figure 4: 42 vertices, point symmetry
|P10, P 41|=|P42, P 31| ≈ 1.036,|P41, P 37|=|P17, P 42| ≈ 1.167
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Figure 5: 42 vertices, red edges = 1 + ε
Even if it contradicts the intuition, Figure 2 (right-sided) and Figure 4 do not show matchstick
graphs which contain vertices of degree 3, 4, and 6. They show 4-regular planar graphs which are not
matchstick graphs.
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3 Examples of 4-regular planar graphs with 50 – 62 vertices that look
like matchstick graphs
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Figure 6: 50 vertices, asymmetric
|P46, P 16| ≈ 1.0797549592,|P48, P 50| ≈ 1.2721354299,|P49, P 50| ≈ 1.2440648255
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Figure 7: 51 vertices, asymmetric
|P42, P 22| ≈ 0.99527617909,|P43, P 4| ≈ 0.99277039021,|P45, P 6| ≈ 1.00583136108
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Figure 8: 51 vertices, asymmetric
|P43,38| ≈ 1.0096420153,|P44, P 40| ≈ 0.9924715954,|P51, P 46| ≈ 1.0027811544
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Figure 9: 51 vertices, asymmetric
|P43, P 4| ≈ 0.9890758621,|P45, P 6| ≈ 1.0027078452,|P46, P 14| ≈ 0.9922899189
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Figure 10: |P48, P 10| ≈ 1.0000027505,|P43, P 38| ≈ 1.0019150342,|P42, P 22| ≈ 0.9887895316
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Figure 11: |P45, P 3| ≈ 0.9896643320,|P45, P 6| ≈ 1.0065187004,|P51, P 50| ≈ 1.0146695273
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Figure 12: |P44, P 40| ≈ 1.0912339657,|P47, P 16| ≈ 1.0142532443,|P50, P 49| ≈ 1.0908924633
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Figure 13: 52 vertices, asymmetric
|P4, P 54| ≈ 0.9029654473,|P38, P 54| ≈ 1.0775714256,|P42, P 41| ≈ 1.0445005488
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Figure 14: |P49, P 43| ≈ 1.0131542972,|P52, P 45| ≈ 0.9838288206,|P52, P 47| ≈ 1.0196014610
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Figure 15: |P27, P 52| ≈ 1.0818208359,|P27, P 53| ≈ 1.0818208359
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Figure 16: |P19, P 53| ≈ 0.9903987194,|P44, P 54| ≈ 0.9903987194
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Figure 17: |P11, P 20| ≈ 0.9862260008,|P45, P 37| ≈ 0.9862260008
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Figure 18: |P49, P 54| ≈ 1.0587135610,|P50, P 53| ≈ 1.0587135610
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Figure 19: |P13, P 40| ≈ 1.0664925618,|P27, P 53| ≈ 1.0664925618
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Figure 20: |P19, P 52| ≈ 1.0897807877,|P51, P 44| ≈ 1.0897807877
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Figure 21: |P16, P 55| ≈ 1.1680810280,|P43, P 55| ≈ 1.1680810280
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Figure 22: |P28, P 24| ≈ 0.9974661859,|P51, P 55| ≈ 0.9974661859
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Figure 23: |P16, P 53| ≈ 0.9944318817,|P44, P 25| ≈ 0.9944318817
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Figure 24: |P6, P 21| ≈ 1.0160374540,|P34, P 49| ≈ 1.0160374540
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Figure 25: |P13, P 53| ≈ 1.0534322768,|P54, P 39| ≈ 1.0534322768
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Figure 26: |P49, P 56| ≈ 1.1077418725,|P50, P 54| ≈ 1.1077418725
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Figure 27: |P25, P 55| ≈ 1.0142467355,|P26, P 54| ≈ 1.0142467355,|P49, P 56| ≈ 1.0142467355,
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Figure 28: |P46, P 57| ≈ 1.0126823415,|P55, P 44| ≈ 1.0126823415,|P56, P 48| ≈ 1.0126823415
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Figure 29: |P50, P 57| ≈ 1.0510811050,|P51, P 55| ≈ 1.0510811050,|P56, P 49| ≈ 1.0510811050
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Figure 30: |P29, P 30| ≈ 1.0171763772,|P57, P 58| ≈ 1.0171763772
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Figure 31: |P29, P 57| ≈ 1.0314677815,|P56, P 58| ≈ 1.0314677815
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Figure 32: |P55, P 58| ≈ 0.9049703660,|P56, P 57| ≈ 0.9049703660
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Figure 33: |P56, P 58| ≈ 1.0319785178,|P57, P 29| ≈ 1.0319785178,|P59, P 57| ≈ 1.1635893883,
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Figure 34: 59 vertices, asymmetric
|P50, P 59| ≈ 1.0143213484,|P56, P 51| ≈ 1.0184958217,|P56, P 58| ≈ 1.0273592144
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Figure 35: 60 vertices, rotational symmetry of order 3
|P58, P 60|=|P58, P 59|=|P59, P 60| ≈ 1.0889437642
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Figure 36: 60 vertices, point symmetry
|P29, P 59|=|P30, P 58| ≈ 0.9013368132
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Figure 37: 61 vertices, mirror symmetry
|P31, P 60| ≈ 0.8891556455,|P31, P 61|=|P60, P 61| ≈ 1.0097449640
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Figure 38: 61 vertices, point symmetry
|P27, P 29|=|P56, P 58| ≈ 1.0396024985,|P60, P 61| ≈ 0.9805271527
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Figure 39: 62 vertices, point symmetry
|P59, P 62|=|P61, P 60| ≈ 1.3001527612
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Figure 40: 62 vertices, point symmetry
|P25, P 62|=|P56, P 32| ≈ 1.0332092374
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Figure 41: 62 vertices, point symmetry
|P59, P 62|=|P60, P 61| ≈ 1.0420956616
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Figure 42: 62 vertices, mirror symmetry
|P61, P 27|=|P62, P 56| ≈ 1.0055442035,|P61, P 62| ≈ 1.3434011015
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Figure 43: 62 vertices, mirror symmetry
|P28, P 61|=|P61, P 56| ≈ 1.0758636209,|P30, P 60|=|P60, P 58| ≈ 0.9760901087
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4 How to find minimal 4-regular matchstick graphs
We are interested in all new solutions on 4-regular matchstick graphs. If you find a new graph, a
better approximate solution for an already existing graph, or a proof, please submit it to the author’s
institutional E-Mail address or website contact.
Here we give a brief instruction for the interested reader how minimal 4-regular matchstick graphs
could be found.
Take a close look at all graphs found so far [3]. Study the design from the outer circle to the
center. Identify rigid and flexible subgraphs (triangles, rhombuses, etc.).
First construct the frame of the graph using only equilateral triangles. Then built bigger rigid
subgraphs of 1, 3, 5, or 7 triangles. Construct the graph from outside to inside. It is always
helpful using a mirror or point symmetry for the frame. Asymmetric graphs are the most difficult
types to handle.
Use the free software MATC HSTI CK GRA PH S CAL CU LATOR (MGC) [2] available on the au-
thor’s website. The MGC contains a brief description of the construction language and a manual
for using the function buttons.
Use a CAD software. All graphs from this article and many others graphs are included into the
MGC and can be downloaded as DXF, SVG, and some other files.
Experiment with already existing graphs. Use subgraphs for creating new graphs.
Do not use matchsticks. Even they give the name of such graphs, matchsticks are the worst
possible method to construct such graphs. They are simply too inaccurate.
Better use flexi filing strip fasteners. Some of the author’s graphs have been found with this
kind of model. Pictures can be found in [7].
Be creative and develop new methods. Perhaps automated methods using artificial intelligence
are the key to success.
24
5 References
[1] Heiko Harborth, Match Sticks in the Plane, The Lighter Side of Mathematics. Proceedings of the
Eug`ene Strens Memorial Conference of Recreational Mathematics & its History, Calgary, Canada,
July 27 – August 2, 1986 (Washington) (Richard K. Guy and Robert E. Woodrow, eds.), Spectrum
Series, The Mathematical Association of America, 1994, pp. 281 – 288.
[2] Stefan Vogel, Matchstick Graphs Calculator (MGC), a software for the construction and calcula-
tion of unit distance graphs and matchstick graphs, (2016 – 2020).
http://mikematics.de/matchstick-graphs-calculator.htm
[3] Mike Winkler, A catalogue of 4-regular matchstick graphs with 63 – 70 vertices and (2; 4)-
regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2,
May 2017.
https://arxiv.org/abs/1705.04715
[4] Mike Winkler, Peter Dinkelacker, and Stefan Vogel, New minimal (4; n)-regular matchstick
graphs, Geombinatorics Volume XXVII, Issue 1, July 2017, Pages 26 – 44.
https://arxiv.org/abs/1604.07134
[5] Mike Winkler, Peter Dinkelacker, and Stefan Vogel, On the existence of 4-regular matchstick
graphs, April 2017.
https://arxiv.org/abs/1705.00293
[6] Mike Winkler, Peter Dinkelacker, and Stefan Vogel, Streichholzgraphen 4-regul¨
ar und 4/n-regul¨
ar
(n>4) und 2/5, thread in a graph theory internet forum, used nicknames: P. Dinkelacker (haribo),
M. Winkler (Slash).
https://tinyurl.com/y3tywma8
[7] Mike Winkler and Stefan Vogel, 4-regul¨
arer Streichholzgraph mit 88 Kanten und 44 Knoten,
thread in a graph theory internet forum, used nickname: M. Winkler (Slash).
https://tinyurl.com/y653xrxx
25
ResearchGate has not been able to resolve any citations for this publication.
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, 57 and 60 vertices. It is shown that for all number of vertices ≥ 63 at least one example of a 4-regular matchstick graph exists.
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 number of vertices.
Matchstick Graphs Calculator (MGC), a software for the construction and calculation of unit distance graphs and matchstick graphs
  • Stefan Vogel
Stefan Vogel, Matchstick Graphs Calculator (MGC), a software for the construction and calculation of unit distance graphs and matchstick graphs, (2016 -2020).
A catalogue of 4-regular matchstick graphs with 63 -70 vertices and (2; 4)-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2
  • Mike Winkler
Mike Winkler, A catalogue of 4-regular matchstick graphs with 63 -70 vertices and (2; 4)-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2, May 2017. https://arxiv.org/abs/1705.04715
4-regulärer Streichholzgraph mit 88 Kanten und 44 Knoten, thread in a graph theory internet forum
  • Mike Winkler
  • Stefan Vogel
Mike Winkler and Stefan Vogel, 4-regulärer Streichholzgraph mit 88 Kanten und 44 Knoten, thread in a graph theory internet forum, used nickname: M. Winkler (Slash). https://tinyurl.com/y653xrxx