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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,

|P50, P 53| ≈ 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