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A 3-Regular Matchstick Graph of Girth 5 Consisting of 54 Vertices

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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.
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arXiv:1903.04304v2 [math.CO] 8 Nov 2019
A 3-REGULAR M ATCHSTI CK GRAPH O F GIRTH 5
CONSISTING OF 54 VERTICES
Mike WinklerPeter Dinkelacker⋆⋆ Stefan Vogel⋆⋆⋆
Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum, Germany, mike.winkler@ruhr-uni-bochum.de
⋆⋆Togostr. 79, 13351 Berlin, Germany, peter@grity.de
⋆⋆⋆Raun, Dorfstr. 7, 08648 Bad Brambach, Germany, backebackekuchen16@gmail.com
November 5, 2019
Abstract
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.
INT RO DUCTION
A 3-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 3.
The girth of a graph is the length of a shortest cycle contained in the graph. Therefore, no rigid
matchstick graph of girth 4exists, because such graphs contain only flexible subgraphs.
In 2010 Sascha Kurz and Giuseppe Mazzuoccolo proved that a 3-regular matchstick graph of
girth 5 consists at least of 30 vertices and gave an example consisting of 180 vertices [1].
In this paper we construct an example with 54 vertices (see Figure 1) and prove its geometrical
correctness. This graph was rst presented by the authors on February 8, 2019 in a graph theory
internet forum [4]. The graph and its proof by construction can also be viewed on the authors web-
site mikematics.de1with a software called MATCHSTI CK GR APHS CALCULATOR (MGC) [2]. This
software runs directly in web browsers.2The MGC includes an animation function for representing
the movement we are mentioning in our proof.
We also found further 3-regular matchstick graphs of girth 5 with less than 70 vertices consisting
of 58, 60, 64, 66, and 68 vertices. These graphs are included in the MGC and exhibit in a separate
paper [5].
1http://mikematics.de/matchstick-graphs-calculator.htm
2For optimal functionality and design please use the Firefox web browser.
Winkler, Dinkelacker, Vogel - A 3-regular matchstick graph of girth 5 consisting of 54 vertices
THE CONSTRUCT ION OF T HE GRA PH
Theorem. A 3-regular matchstick graph of girth 5 consisting of 54 vertices exists.
Proof. Given are the vertices P1and P2with the distance of a unit length in the plane. In the
following an ’edge’ means always an ’edge of unit length’. Add an edge from P2to P1. Add an
edge from P3to P1with angle α= 102to the edge from P2to P1. Add an edge from P4to P3
with angle β= 67to the edge from P1to P3. Add an edge from P5to P3with angle γ= 74
to the edge from P4to P3. Add an edge from P6to P5with angle δ= 81to the edge from
P3to P5. Add two edges to complete the isosceles triangle P7, P 2, P 4, the base remains open.
Add two edges to complete the isosceles triangle P8, P 4, P 6, the base remains open. Add an edge
from P9to P8with angle ǫ= 24to the edge from P4to P8. Add two edges to complete the
isosceles triangle P10, P 9, P 7, the base remains open. Add an edge from P11 to P5with angle
ζ= 69to the edge from P6to P5. Add an edge from P12 to P11 with angle η= 106to the
edge from P5to P11. Add two edges to complete the isosceles triangle P13, P 6, P 12, the base
remains open. Add two edges to complete the isosceles triangle P14, P 13, P 9, the base remains
open. Add an edge from P15 to P14 with angle θ= 3to the edge from P9to P14. Add an
edge from P16 to P11 with angle ι= 61to the edge from P12 to P11. Add two edges to
complete the isosceles triangle P17, P 12, P 15, the base remains open. Add two edges to complete
the isosceles triangle P18, P 17, P 16, the base remains open. Add an edge from P19 to P1with
angle κ= 85to the edge from P2to P1. Add an edge from P20 to P19 with angle λ= 91to
the edge from P1to P19. Add two edges to complete the isosceles triangle P21, P 20, P 2, the base
remains open. Add two edges to complete the isosceles triangle P22, P 10, P 21, the base remains
open. Add two edges to complete the isosceles triangle P23, P 15, P 22, the base remains open.
Add two edges to complete the isosceles triangle P24, P 18, P 23, the base remains open. Add an
edge from P25 to P19 with angle µ= 38to the edge from P20 to P19. Add an edge from P26
to P16 with angle ν= 65to the edge from P18 to P16. Add two edges to complete the isosceles
triangle P27, P 24, P 26, the base remains open. Connect P25 and P26 by a copy of the subgraph
P1P27. The copied subgraph matches the already existing graph, because of congruence. Add
two edges to complete the isosceles triangle P53, P 47, P 27, the base remains open. Add two edges
to complete the isosceles triangle P54, P 20, P 52, the base remains open. Add an edge of unknown
length from P53 to P54.
We get a point-symmetric 3-regular planar graph of girth 5 consisting of 54 vertices. The
graph contains clearly visible distances between vertices and edges (see Figure 1). The method
of construction ensures that each edge, except P53, P 54, has unit length. The distance P53, P 54
measures 1.0007. Varying the angle µhas no effect on the other twelve angles mentioned before.
Because of the point symmetry we can separate the graph into two rigid subgraphs G1and G2.G1
consisting of P1P24, P 26, P 27, and G2consisting of P25, P 28 P52. By holding G1fixed
and changing µcontinuously close to its value from about 39to 37, the distance P53, P 54 varies
around the unit length from approximately 0,991 to 1,012. During this changing G2moves slightly
from the right to the left, while P53 moves up and P54 moves down. During the movement there
are no unauthorized overlaps or contacts. This shows that a 3-regular matchstick graph of girth 5
consisting of 54 vertices exists. The distance P53, P 54 has unit length, if µ38,067338069376.
2
Winkler, Dinkelacker, Vogel - A 3-regular matchstick graph of girth 5 consisting of 54 vertices
α
β
γ
δ
ǫ
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η
θ
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P1
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P3
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P17
P18
P19
P20
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P24 P25
P26
P27
P28
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P30
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P42 P43
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P54
Figure 1: Only known example of a 3-regular matchstick graph of girth 5 consisting of 54 vertices.
OPE N PROBLEMS AN D CONJ ECTURES A BOUT
MATCHSTI CK GRA PHS
Kurz and Mazzuoccolo write in their publication of 2010: ”Our knowledge about matchstick
graphs is still very limited. It seems to be hard to obtain rigid mathematical results about them.
[1]” Unfortunately, nearly a decade later this statement is still valid. One of the reasons for this,
besides the mathematically difficulty, is certainly that still too few people deal with this topic. We
hope this article could change this. The following list of open problems and conjectures should be
an incentive for the interested reader.
Problem 1: Does a 3-regular matchstick graph of girth 5 with less than 54 vertices exists?
Problem 2: 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 [3]. Try to find an example for one of the missing
3
Winkler, Dinkelacker, Vogel - A 3-regular matchstick graph of girth 5 consisting of 54 vertices
numbers of vertices, especially with less than 52 vertices.
Problem 3: Is there an elegant proof or an algorithm which is relatively fast in practice to show that
a planar graph with less than 10 vertices can be a matchstick graph?
Conjecture: The so-called ”Harborth-Graph” with 52 vertices is the smallest possible example
of a 4-regular matchstick graph that contains two lines of symmetry, as in a rhombus. A proof of
this conjecture may be possible.
REF ERENCES
[1] Sascha Kurz and Giuseppe Mazzuoccolo, 3-regular matchstick graphs with given
girth, Geombinatorics Quarterly Volume 19, Issue 4, April 2010, pp. 156–175.
https://arxiv.org/abs/1401.4360
[2] Stefan Vogel, Matchstick Graphs Calculator (MGC), a software for the construction and cal-
culation of unit distance graphs and matchstick graphs, (2016–2019).
http://mikematics.de/matchstick-graphs-calculator.htm
[3] Mike Winkler, Peter Dinkelacker, and Stefan Vogel, On the Existence of 4-regular Matchstick
Graphs, June 2017. https://arxiv.org/abs/1705.00293
[4] 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, post No.1698, P. Dinkelacker
(haribo), M. Winkler (Slash). http://tinyurl.com/y2em6k9o
[5] Mike Winkler, A catalog of 3-regular matchstick graphs of girth 5 consisting of 54 – 68 vertices,
March 2019. http://mikematics.de/catalog-3reg-girth5.pdf
4
Preprint
Full-text available
This article exhibits the currently smallest known examples of 3-regular matchstick graphs of girth 5 consisting of less than 70 vertices.
Article
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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.
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We consider 3-regular planar matchstick graphs, i.e. those which have a planar embedding such that all edge lengths are equal, with given girth g. For girth 3 it is known that such graphs exist if and only if the number of vertices n is an even integer larger or equal to 8. Here we prove that such graphs exist for girth g=4 if and only if n is even and at least 20. We provide an example for girth g=5 consisting of 180 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, (2016-2019).
Streichholzgraphen 4-regulär und 4/nregulär (n>4) und 2/5, thread in a graph theory internet forum
  • Mike Winkler
  • Peter Dinkelacker
  • Stefan Vogel
Mike Winkler, Peter Dinkelacker, and Stefan Vogel, Streichholzgraphen 4-regulär und 4/nregulär (n>4) und 2/5, thread in a graph theory internet forum, post No.1698, P. Dinkelacker (haribo), M. Winkler (Slash). http://tinyurl.com/y2em6k9o
A catalog of 3-regular matchstick graphs of girth 5 consisting of 54 -68 vertices
  • Mike Winkler
Mike Winkler, A catalog of 3-regular matchstick graphs of girth 5 consisting of 54 -68 vertices, March 2019. http://mikematics.de/catalog-3reg-girth5.pdf