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4-regular planar unit triangle graphs

without additional triangles

Mike Winkler1Peter Dinkelacker2Stefan Vogel3

1Fakult¨

at f¨

ur Mathematik, Ruhr-Universit¨

at Bochum, Germany,

mike.winkler@ruhr-uni-bochum.de

2Togostr. 79, 13351 Berlin, Germany, peter@grity.de

3Raun, Dorfstr. 7, 08648 Bad Brambach, Germany,

backebackekuchen16@gmail.com

Abstract

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.

1Introduction

In 1991 it was shown by Harborth[1] that the smallest number of non-

overlapping vertex-to-vertex unit triangles in the plane is ≤42 in general

(see Figure 1), and ≤3800 if additional triangles are not allowed (see

Figure 2). The smallest example of an additional triangle is an equilateral

triangle consisting of three unit triangles. The graphs mentioned by Har-

borth are not planar graphs, because vertices lie exactly on other edges.

So it remained an open question, whether 4-regular planar unit-distance

graphs consisting only of unit triangles without additional triangles exists.

In this paper we prove the existence of such graphs. The currently small-

est known example consists of 6422 unit triangles and has a rotational

symmetry of order 192.

1

arXiv:1902.00966v2 [math.MG] 3 Nov 2019

Figure 1 shows the only known examples of 4-regular planar unit-

distance graphs with 42 unit triangles. Both graphs are rigid and con-

structed from six copies of the same rigid subgraph (gray). The equilat-

eral triangle (A, B, C )shows one of the additional triangles.

A

BC

Figure 1

For graphs without additional triangles Harborth presented the sub-

graph G1which consists of 38 unit triangles (see Figure 2).

γγγγγ

β

αααα

g1

g2

A

B

CD

E

F

G H

γγγγγ

β

αααα

g1

g2

A

B

CD

E

F

G H

γγγγγ

β

αααα

g1

g2

A

B

CD

E

F

G H

γγγγγ

β

αααα

g1

g2

A

B

CD

E

F

G H

γγγγγ

β

αααα

g1

g2

A

B

CD

E

F

G H

Figure 2: The non-planar subgraph G1.

2

For a proof of the non-planarity of G1we refer the reader to [1]. The

proof for its existence is similar to the proof of the main Theorem.

2The main Theorem

For the main Theorem we use the planar subgraph G2which also con-

sists of 38 unit triangles (see Figure 3). The graph achieves its planarity

through a slight modiﬁcation of the right side.

γ γ γ

β

ααα

δ

g1

g2

A

B

CD

E

F

G H

γ γ γ

β

ααα

δ

g1

g2

A

B

CD

E

F

G H

γ γ γ

β

ααα

δ

g1

g2

A

B

CD

E

F

G H

γ γ γ

β

ααα

δ

g1

g2

A

B

CD

E

F

G H

γ γ γ

β

ααα

δ

g1

g2

A

B

CD

E

F

G H

Figure 3: The planar subgraph G2.

Theorem. The number of unit triangles in 4-regular planar unit-

distance graphs consisting only of unit triangles without additional tri-

angles is ≤6422.

Proof. We arrange 38 unit triangles in the plane in such a way that

we get the graph G2which only contains vertices of degree 2 and 4. The

vertices of degree 2 lie exactly on the two intersecting lines g1and g2,

where the point of intersection is right of E(see Figure 3). G2is ﬂexible

3

and mirror-symmetric about BE. Let nbe an integer and let ω=360◦

n

be the angle between g1and g2. Let us imagine g1and g2as rails on

which the vertices with the exception of Dand Fare movably mounted.

Then, because of its ﬂexibility, G2can be pushed back and forth slightly,

whereby Dand Fmove up and down. With such a movement no angle

in the graph remains constant. If one of the angles α,β, or γis changed

continuously close to its value given in Table 1, then the path of Dand F

sometime cuts g1and g2. This intersection can not be speciﬁed exactly,

but it must exist. During this movement of Dand Facross g1and g2it

must be ensured that each vertex of degree 4 do not intersect with other

edges nor with g1and g2. The smallest integer nfor which this applies

is n= 169. Because of the mirror symmetry we can connect 169 copies

of G2together in such a way that the vertices Aand C, as well as F

and Dcoincide to build a rigid planar 4-regular ring graph consisting of

169 ·38 = 6422 unit triangles. The vertex of ωis the center of this ring.

Because G1is also mirror-symmetric about BE, the proof for the ex-

istence of a rigid ring graph consisting of 100 copies of G1is similar to

the proof before. Table 1 gives the approximate values for the angles and

distances in both subgraphs.

G1G2

](−→

F A, −−→

DC) = 360◦/100 = 3,6◦360◦/169 ≈2,130◦

α≈91,58566772584003◦78,95050838942406◦

β≈43,94364026236698◦38,40835335322197◦

γ≈119,90161279889431◦119,99637583181277◦

δ≈–122,42510282308054◦

GH ≈0,00171718039014 0,00006325366750

Table 1

4

There exists also a 4-regular planar unit triangle graph with a rota-

tional symmetry of order 100. This rigid ring graph consists of 9200 unit

triangles and is build from the subgraph G3(see Figure 4). As well as G2,

these graph contains very small distances between vertices and edges.

Figure 4: The planar subgraph G3.

3Other forms besides rings

In G1the vertices on BE are mirror-symmetric about g3, the perpendicu-

lar bisector of BE. Let G4be the graph we get by reﬂecting each vertex

beneath BE over g3, by which g1and g2become parallel (see Figure 5).

Let G5be the graph we get by reﬂecting each vertex of G4over BE.

Now we can use G4and G5as a kind of adapters to change the direction

of the ring segments build with G1, which makes inﬁnitely many other

forms besides ring graphs possible (see Figure 6). But, as well as G1, all

these graphs are not planar. It remains an open question, if also planar

unit triangle graphs in other forms besides rings exist.

γγγγγ

β

αααα

g1

g2

g3

A

B

CD

E

F

G H

Figure 5: The non-planar subgraph G4.

5

Figure 6: Non-planar examples of other forms besides ring graphs.

Each graph from this article, as well as the ring graphs, can be viewed

on the authors website mikematics.de1. The software[2] we also used to

verify the graphs runs directly in web browsers2. The results of this article

were ﬁrst presented by the authors between September 17 and November

3, 2018 in a graph theory internet forum[3].

4References

[1] Heiko Harborth, Plane four-regular graphs with vertex-to-vertex unit

triangles, Discrete Mathematics 97 (1991) pp. 219–222.

[2] Stefan Vogel, Matchstick Graphs Calculator (MGC), a software for

the construction and calculation of unit distance graphs and match-

stick graphs, (2016–2019).

http://mikematics.de/matchstick-graphs-calculator.htm

[3] Mike Winkler, Peter Dinkelacker, and Stefan Vogel, Streichholz-

graphen 4-regul¨

ar und 4/n-regul¨

ar (n>4) und 2/5, thread in a graph

theory internet forum, posts No.1412–1585, P. Dinkelacker (haribo),

M. Winkler (Slash), https://tinyurl.com/yaun9yrn

1http://mikematics.de/matchstick-graphs-calculator.htm

2For optimal functionality and design please use the Firefox web browser.

6