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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 Winkler⋆Peter 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 ﬁrst 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= 102◦to the edge from P2to P1. Add an edge from P4to P3

with angle β= 67◦to 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= 81◦to 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= 24◦to 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= 69◦to the edge from P6to P5. Add an edge from P12 to P11 with angle η= 106◦to 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= 3◦to the edge from P9to P14. Add an

edge from P16 to P11 with angle ι= 61◦to 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= 85◦to the edge from P2to P1. Add an edge from P20 to P19 with angle λ= 91◦to

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= 38◦to the edge from P20 to P19. Add an edge from P26

to P16 with angle ν= 65◦to 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

P1−P27. 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 P1−P24, P 26, P 27, and G2consisting of P25, P 28 −P52. By holding G1ﬁxed

and changing µcontinuously close to its value from about 39◦to 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 µ≈38,067338069376◦.

2

Winkler, Dinkelacker, Vogel - A 3-regular matchstick graph of girth 5 consisting of 54 vertices

α

β

γ

δ

ǫ

ζ

η

θ

ι

κ

λ

µ

ν

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

P16

P17

P18

P19

P20

P21

P22

P23

P24 P25

P26

P27

P28

P29

P30

P31

P32

P33

P34

P35

P36

P37

P38

P39

P40

P41

P42 P43

P44

P45

P46

P47

P48

P49

P50

P51

P52

P53

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

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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

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