Crossing Number Graphs »

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The Mathematica

®

Journal

B E Y O N D S U D O K U

Crossing Number Graphs

Ed Pegg Jr and Geoffrey Exoo

Sudoku is just one of hundreds of great puzzle types. This column

pre-sents obscure logic puzzles of various sorts and challenges the readers to

solve the puzzles in two ways: by hand and with

Mathematica

. For the

lat-ter, solvers are invited to send their code to [email protected]. The

per-son submitting the most elegant solution will receive a prize.

‡

The Brick Factory

In 1940, the Hungarian mathematician Paul Turán was sent to a forced labor

camp by the Nazis. Though every part of his life was brutally controlled, he still

managed to do serious mathematics under the most extreme conditions. While

forced to collect wire from former neighborhoods, he would be thinking about

mathematics. When he found scraps of paper, he wrote down his theorems and

conjectures. Some of these scraps were smuggled to Paul Erd

ő

s. One problem he

developed is now called the brick factory problem.

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‡

Complete Graphs

Turán built the field of extremal graph theory while a prisoner. Many good

puz-zles come from graphs, such as the familiar problem of connecting water, gas,

and electricity to three houses without the lines crossing. This would be

equiva-lent to

K

3,3

or 3 kilns, 3 storage yards. Without some form of cheating, the

prob-lem is impossible, since one crossing is required. (Try proving the impossibility

with Euler’s formula: vertices + faces - edges = 2). Consider the brick factory

problem with 4 kilns at the top, and 3 storage yards at the bottom, or

K

3,4

. What

is the minimal number of crossings? If you are looking at the notebook version

of this column in

Mathematica

6 or later, you can drag the vertices

~

first

double-click a vertex twice.

CompleteBipartite,3, 4

For

K

_7,7

, in 1970 Kleitman proved the minimal number of crossings was 77, 79,

or 81 [2]. He ended his paper with “Which?”

~

a question not solved until 1993,

when Woodall showed the answer was 81 [3].

For many years, the number of crossings for the complete graph

K

10

was known

to be either 61 or 62, with a proof of 62 finally arriving in 2000 [4]. Below are

pictures of the graph in two forms, one symmetric, the other with minimal

crossings. Due to the tediousness of counting that many crossings by hand, and

the 40-plus years required to solve it, I do not consider this a friendly puzzle for

casual solving. The Rectilinear Crossing Number (RCN) project, maintained by

Oswin Aichholzer, has solved all complete graphs up to

K

19

, with 1318 crossings

[5]. The calculation and bounding of crossing numbers remains active to this day

[6, 7].

Complete, 10

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‡

GraphData

is an extensive built-in data paclet included with

Mathematica

7 [8]. It

contains more than 3000 graphs, in excess of 200 graph properties, and one or

more attractive embeddings for most of its constituent graphs.

GraphData

has

been painstakingly compiled by Eric Weisstein and myself making use of

exten-sive

Mathematica

computations, programming, hand-drawings, careful mining

of the graph theoretic literature, and feedback from a number of prominent

graph theorists. (If you have any additions to suggest for

GraphData

, please feel

free to email them to [email protected].) For example, here are a few 30-vertex

graphs from

GraphData[30]

.

CubicTransitive, 57 FosterCage LeviGraph

DoubleStarSnark MeringerGraph WongGraph

‡

Crossing Number Graphs

A graph with no crossings is a planar graph. Here is a Wolfram Demonstration

(“Planarity”) for solving planar graph embedding puzzles [9, 10]. You can drag

vertices.

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random coordinates

choose graph 4

One piece of research I did, with substantial help from Geoffrey Exoo [11], was

to catalog the smallest cubic graphs that require a certain number of crossings. I

named these objects “crossing number graphs”, or CNGs in this column. We

have already met CNG 1, since

K

3,3

requires one crossing. With two crossings,

there are two minimal cubic graphs, one the famous Petersen graph.

PetersenGraph _{CNG 2B}

With the 3-crossing graphs, there are eight graphs to choose from, with two

of them being famous. CNG 3E has been shown in a 3-crossing form; the rest

are given as embedding puzzles.

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CNG 3A CNG 3B Heawood graph CNG 3D

CNG 3E CNG 3F GP7,2 CNG 3H

In 4-crossing graphs, both are famous. The crossed-prism graph has been solved

for you. The 4-crossing version of the Möbius

|

Kantor graph can probably be

im-proved. This comment applies to almost all embeddings in this column.

MöbiusKantor graph 8crossed prism graph

With 5-crossing cubic graphs, one is famous. So far, all the crossing number

graphs included a famous graph, and I thought I was seeing a trend. Both

of these are difficult embedding puzzles.

Pappus graph CNG 5B

At 6 crossings, all three graphs were incidence graphs for 10

3

configurations.

Configuration puzzle: arrange 10 points to make 10 lines of three points, with

three lines through each point. There are 10 such configurations [12]. Again, one

famous graph.

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Desargues graph 1032incidence 1036incidence

The trend of crossing number graphs being famous was shattered with the

7-crossing graphs. There are four such graphs, and none of them seems to be

famous.

CNG 7A CNG 7B CNG 7C CNG 7D

The five 8-crossing graphs contain the McGee graph and Foster 24A.

McGeeGraph F024A CNG 8C CNG 8D CNG 8E

With 26 vertices and beyond, the authors make conjectures. With 26, 28, and 30

vertices, the conjectured maximal crossings are 10, 11, and 13. Note that 9

cross-ings would not be extremal. The authors welcome expansion of these results. A

that the crossing number is less than the rectilinear crossing number.

McGeeedge Coxeter graph Tutte 8cage

No other crossing number graphs are known at this time. If you find any nice

embeddings for any of these graphs, please send them to [email protected]. The

crossing number graphs were found by co-author Geoffrey Exoo, using his

pro-gram gnubar (nu,

n

, represents a crossing number; a bar suggests rectilinearity).

The many cubic graphs he tested were obtained from Brendan McKay’s nauty

program [13]. A brief description of the gnubar analysis of 7-crossing graphs is as

follows:

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No other crossing number graphs are known at this time. If you find any nice

embeddings for any of these graphs, please send them to [email protected]. The

crossing number graphs were found by co-author Geoffrey Exoo, using his

pro-gram gnubar (nu,

n

, represents a crossing number; a bar suggests rectilinearity).

The many cubic graphs he tested were obtained from Brendan McKay’s nauty

program [13]. A brief description of the gnubar analysis of 7-crossing graphs is as

follows:

1. Set as the target the highest crossing number possible. So far, adding

two vertices increases the maximum crossing number by one.

2. Run a fast drawing routine that looks for reasonably good drawings for

each graph. An embedding with fewer crossings than the target number

eliminates that graph. For

n

=

22, out of 1435720 graphs, 50 graphs

remained.

3. A different randomized graph embedding routine found drawings with

six or fewer crossings for all but the four graphs.

4. Prove that there are no better drawings for these four graphs. This step

takes the most time

~

roughly 150 hours per graph.

‡

Solution for Number Link (Issue 10:3)

· Recap of Number Link Rules

As mentioned in Volume 10, Issue 3, a Number Link puzzle follows four basic

rules:

1. A continuous line must connect identical numbers.

2. Lines must go through the center of a cell horizontally or vertically and

never go through the same cell twice.

3. Lines cannot cross, branch off, or go through other numbered cells.

4. Every unnumbered square must contain part of a path.

The only person sending in a solving method was Don Knuth.

“Has anybody to your knowledge pointed out that there’s an easy way

to “reduce” Number Link to finding Hamiltonian paths? Namely,

suppose there are four links, with the digit 1 in cells {1a,1b}, ... , and

digit 4 in cells {4a,4b}. Extend the

n

µ

n

grid graph by adding four

new vertices [1], [2], [3], [4], and eight new edges 1b

~

[1]

~

2a, 2b

~

[2]

~

3a, 3b

~

[3]

~

4a, 4b

~

[4]

~

1a. A solution to the original puzzle

is equivalent to a Hamiltonian path in this new graph.”

· Don Knuth’s Method

First, for one of the puzzles, five extra points are added to make the base graph.

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In[47]:= pts1, 1,3, 6,3, 2,1, 3,

6, 2,6, 7,2, 1,7, 7,5, 3,6, 6; graph  FlattenSelectSubsetsTuplesRange7,2,2,

Norm1 21 &,

FlattenMapIndexed8,212, 1&,

PartitionRotateLeftFlattenpts, 1, 1, 2,2, 1, 1; Graphics

MapIndexed

TextStyle21, FontSize 14,1&,pts12,2, FirstGraphPlotRulegraph,

VertexCoordinateRulesThreadUnionFlattengraph, 1 UnionFlattengraph, 1Table.5, .5,54, GridLines Range0, 8, Range0, 8, PlotRange

0, 8,0, 7, ImageSize120, 100

Out[49]=

1

2

3

4

5

5 After that, all Hamiltonian cycles for the graph are found

~

66648 different

cy-cles in total. The desired graph has the defining point set in order.

In[50]:= GraphUtilities`

hamcyc HamiltonianCyclesRulegraph, 100 000; Lengthhamcyc

SelectRangeLengthhamcyc,

Selecthamcyc, MemberQFlattenpts, 1,& Flattenpts, 1&

Out[52]= 66 648

Out[53]= 49 823

From that, the solution immediately follows.

(9)

In[54]:= solution  SelectPartitionhamcyc49 823, 2, 1, 1, NotMemberQFlatten, 8&;

Graphics MapIndexed

TextStyle21, FontSize14,1&,pts12,2, FirstGraphPlotRulesolution,

VertexCoordinateRulesThreadUnionFlattensolution, 1

UnionFlattensolution, 1Table.5, .5,49, GridLines Range0, 7, Range0, 7, PlotRange

0, 7,0, 7, ImageSize120, 100

Out[55]=

1

2

3

4

5

5 For solving, Don Knuth has received the

Nikoli Puzzle Cyclopedia

.

‡

References

[1] P. Turán, “A Note of Welcome,” Journal of Graph Theory, 1(1), 1977 pp. 7|9.

[2] D. J. Kleitman, “The Crossing Number of K5,n,” Journal of Combinatorial Theory, 9, 1970 pp. 315|323.

[3] D. R. Woodall, “Cyclic-Order Graphs and Zarankiewicz’s Crossing-Number Conjecture,” Journal of Graph Theory, 17(6), 1993 pp. 657|691. doi.10.1002/jgt.3190170602.

[4] A. Brodsky, S. Durocher, and E. Gethner. “The Rectilinear Crossing Number of K10Is 62.”

(Sep 22, 2000) arxiv.org/abs/cs/0009023.

[5] O. Aichholzer. “On the Rectilinear Crossing Number.” (2006) www.ist.tugraz.at/staff/ aichholzer/research/rp/triangulations/crossing.

[6] E. de Klerk, J. Maharry, D. V. Pasechnik, R. B. Richter, and G. Salazar. “Improved Bounds for the Crossing Numbers of Km,n and Kn.” (Apr 6, 2004) arxiv.org/abs/math/0404142.

[7] “GraphData” from the Wolfram Mathematica Documentation Center. reference.wolfram.com/mathematica/ref/GraphData.html.

[8] J. Pach, J. Spencer, and G. Tóth, “New Bounds on Crossing Numbers,” in Proceedings of the Fifteenth Annual Symposium on Computational Geometry (SCG'99), Miami Beach, FL (V. Milenkovic, ed.), New York: ACM Press, 1999 pp. 124|133.

citeseer.ist.psu.edu/520124.

[9] E. Pegg Jr. “Planarity” from The Wolfram Demonstrations Project. demonstrations.wolfram.com/Planarity.

[10] E. W. Weisstein. “Planar Graph” from MathWorld~A Wolfram Web Resource. mathworld.wolfram.com/PlanarGraph.

[11] G. Exoo. “Rectilinear Drawings of Famous Graphs.” (2006) ginger.indstate.edu/ge/ Graphs/RECTILINEAR.

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[12] B. Grunbaum, “Lectures on Configurations,” in Proceedings of the International Configu-rations 2004 Workshop, villa Plemelj, Bled, 2004.

[13] B. D. McKay. “The Nauty Page.” (2008) cs.anu.edu.au/~bdm/nauty.

E. Pegg Jr, “Crossing Number Graphs,” The Mathematica Journal, 2011. dx.doi.org/doi:10.3888/tmj.11.2–2.

Pegg

Scientific Information Editor Wolfram Research,

Professor Computer

Haute,

Pegg

Scientific Information Editor Wolfram Research,

Pegg Jr

Scientific Information Editor Wolfram Research, Inc.

Mathematics Computer Science Indiana University

About the Author

Pegg Jr

Scientific Information Editor Wolfram Research, Inc.

Mathematics University Ed Pegg Jr

Scientific Information Editor Wolfram Research, Inc. [email protected]

Geoffrey Exoo

Professor of Mathematics and Computer Science Indiana State University

(11)

Supplement

Initialization

CrossingNumberGraph2A =881Ø3, 1Ø4, 1Ø6, 2Ø4, 2Ø5, 2Ø7, 3Ø5, 3Ø8, 4Ø9, 5Ø10, 6Ø7, 6Ø10, 7Ø8, 8Ø9, 9Ø10<,

81Ø80.`, 1.`<, 2Ø8-0.951`, 0.309`<, 3Ø8-0.588`,-0.809`<, 4Ø80.588`,-0.809`<, 5Ø80.951`, 0.309`<, 6Ø80.`, 2.`<, 7Ø8-1.902`, 0.618`<, 8Ø8-1.176`,-1.618`<,

9Ø81.176`,-1.618`<, 10Ø81.902`, 0.618`<<<;

CrossingNumberGraph2B =881Ø2, 1Ø3, 1Ø7, 2Ø4, 2Ø8, 3Ø5, 3Ø9, 4Ø6, 4Ø10, 5Ø6, 5Ø7, 6Ø8, 7Ø9, 8Ø10, 9Ø10<,

81Ø80.623`,-0.699`<, 2Ø80.623`,-0.137`<, 3Ø80.953`,-1.153`<, 4Ø80.953`, 0.318`<, 5Ø81.487`,-1.327`<, 6Ø81.487`, 0.492`<, 7Ø82.022`,-1.153`<, 8Ø82.022`, 0.318`<, 9Ø82.352`,-0.699`<, 10Ø82.352`,-0.137`<<<;

CrossingNumberGraph3A=

881Ø2, 1Ø5, 1Ø7, 2Ø4, 2Ø6, 3Ø4, 3Ø8, 3Ø12, 4Ø5, 5Ø6, 6Ø9, 7Ø8, 7Ø13, 8Ø9, 9Ø10, 10Ø11, 10Ø14, 11Ø12, 11Ø13, 12Ø14, 13Ø14<,

81Ø80.223`, 0.975`<, 2Ø8-0.223`, 0.975`<,

3Ø8-0.901`,-0.434`<, 4Ø8-1.`, 0.`<, 5Ø8-0.901`, 0.434`<, 6Ø8-0.623`, 0.782`<, 7Ø80.623`, 0.782`<,

8Ø80.901`, 0.434`<, 9Ø81.`, 0.`<, 10Ø80.901`,-0.434`<, 11Ø8-0.223`,-0.975`<, 12Ø8-0.623`,-0.782`<,

13Ø80.223`,-0.975`<, 14Ø80.623`,-0.782`<<<;

CrossingNumberGraph3B=

881Ø2, 1Ø8, 1Ø12, 2Ø3, 2Ø6, 3Ø5, 3Ø7, 4Ø5, 4Ø8, 4Ø12, 5Ø6, 6Ø7, 7Ø11, 8Ø9, 9Ø10, 9Ø14, 10Ø11, 10Ø13, 11Ø14, 12Ø13, 13Ø14<,81Ø80.901`, 0.434`<, 2Ø81.`, 0.`<, 3Ø80.223`,-0.975`<, 4Ø8-0.623`,-0.782`<, 5Ø8-0.223`,-0.975`<, 6Ø80.901`,-0.434`<,

(12)

CrossingNumberGraph3C=

81Ø80.900969`, 0.433884`<, 2Ø80.62349`, 0.781831`<, 3Ø80.222521`, 0.974928`<, 4Ø8-0.222521`, 0.974928`<, 5Ø8-0.62349`, 0.781831`<, 6Ø8-0.900969`, 0.433884`<, 7Ø8-1.`, 0<, 8Ø8-0.900969`,-0.433884`<,

9Ø8-0.62349`,-0.781831`<, 10Ø8-0.222521`,-0.974928`<, 11Ø80.222521`,-0.974928`<, 12Ø80.62349`,-0.781831`<, 13Ø80.900969`,-0.433884`<, 14Ø81.`, 0<<<;

CrossingNumberGraph3D=

881Ø2, 1Ø3, 1Ø6, 2Ø5, 2Ø14, 3Ø4, 3Ø10, 4Ø5, 4Ø7, 5Ø9, 6Ø7, 6Ø8, 7Ø12, 8Ø9, 8Ø13, 9Ø11, 10Ø11, 10Ø13, 11Ø12, 12Ø14, 13Ø14<,81Ø8-0.223`, 0.975`<, 2Ø80.223`, 0.975`<, 3Ø8-0.623`, 0.782`<, 4Ø8-0.901`, 0.434`<, 5Ø8-1.`, 0.`<, 6Ø80.901`,-0.434`<, 7Ø81.`, 0.`<, 8Ø80.623`,-0.782`<, 9Ø8-0.901`,-0.434`<, 10Ø8-0.223`,-0.975`<,

11Ø8-0.623`,-0.782`<, 12Ø80.901`, 0.434`<, 13Ø80.223`,-0.975`<, 14Ø80.623`, 0.782`<<<;

CrossingNumberGraph3E=

81Ø80, 0<, 2Ø80, 3<, 3Ø81, 1<, 4Ø81, 2<, 5Ø82, 0<, 6Ø82, 2<, 7Ø81, 4<, 8Ø83, 1<, 9Ø82, 3<, 10Ø83, 2<, 11Ø83, 3<, 12Ø84, 0<, 13Ø84, 2<, 14Ø84, 4<<<;

CrossingNumberGraph3F=

81Ø80.223`, 0.975`<, 2Ø80.623`, 0.782`<, 3Ø8-0.223`,-0.975`<, 4Ø8-0.623`,-0.782`<, 5Ø8-0.223`, 0.975`<, 6Ø80.223`,-0.975`<, 7Ø8-0.901`,-0.434`<, 8Ø80.901`, 0.434`<, 9Ø8-0.623`, 0.782`<, 10Ø80.623`,-0.782`<, 11Ø8-1.`, 0.`<, 12Ø8-0.901`, 0.434`<, 13Ø81.`, 0.`<, 14Ø80.901`,-0.434`<<<;

(13)

CrossingNumberGraph3G=

881Ø3, 1Ø6, 1Ø8, 2Ø4, 2Ø7, 2Ø9, 3Ø5, 3Ø10, 4Ø6, 4Ø11, 5Ø7, 5Ø12, 6Ø13, 7Ø14, 8Ø9, 8Ø14, 9Ø10, 10Ø11, 11Ø12, 12Ø13, 13Ø14<,81Ø80.623`, 0.782`<, 2Ø8-0.223`, 0.975`<, 3Ø8-0.901`, 0.434`<,

4Ø8-0.901`,-0.434`<, 5Ø8-0.223`,-0.975`<,

6Ø80.623`,-0.782`<, 7Ø81.`, 0.`<, 8Ø81.247`, 1.564`<, 9Ø8-0.445`, 1.95`<, 10Ø8-1.802`, 0.868`<,

11Ø8-1.802`,-0.868`<, 12Ø8-0.445`,-1.95`<, 13Ø81.247`,-1.564`<, 14Ø82.`, 0.`<<<;

CrossingNumberGraph3H=

881Ø2, 1Ø10, 1Ø13, 2Ø5, 2Ø9, 3Ø4, 3Ø6, 3Ø13, 4Ø5, 4Ø8, 5Ø7, 6Ø7, 6Ø10, 7Ø12, 8Ø9, 8Ø11, 9Ø14, 10Ø11, 11Ø12, 12Ø14, 13Ø14<,81Ø80.901`, 0.434`<, 2Ø80.623`, 0.782`<, 3Ø8-0.623`,-0.782`<,

4Ø8-0.223`,-0.975`<, 5Ø80.223`,-0.975`<, 6Ø80.901`,-0.434`<, 7Ø80.623`,-0.782`<,

8Ø8-0.223`, 0.975`<, 9Ø80.223`, 0.975`<, 10Ø81.`, 0.`<, 11Ø8-0.623`, 0.782`<, 12Ø8-0.901`, 0.434`<,

13Ø8-0.901`,-0.434`<, 14Ø8-1.`, 0.`<<<;

881Ø2, 1Ø3, 1Ø6, 2Ø14, 2Ø16, 3Ø4, 3Ø9, 4Ø5, 4Ø8, 5Ø11, 5Ø16, 6Ø7, 6Ø12, 7Ø8, 7Ø10, 8Ø14, 9Ø10, 9Ø15, 10Ø11, 11Ø13, 12Ø13, 12Ø15, 13Ø14, 15Ø16<,

81Ø80, 0<, 2Ø80, 5<, 3Ø81, 1<, 4Ø81, 3<, 5Ø81, 4<, 6Ø82, 0<, 7Ø82, 2<, 8Ø82, 3<, 9Ø83, 1<, 10Ø83, 2<, 11Ø83, 3<, 12Ø84, 0<, 13Ø84, 3<, 14Ø84, 4<, 15Ø85, 1<, 16Ø85, 5<<<;

881Ø2, 1Ø8, 1Ø16, 2Ø3, 2Ø11, 3Ø4, 3Ø10, 4Ø5, 4Ø13, 5Ø6, 5Ø12, 6Ø7, 6Ø15, 7Ø8, 7Ø14, 8Ø9, 9Ø10, 9Ø16, 10Ø11, 11Ø12, 12Ø13, 13Ø14, 14Ø15, 15Ø16<,

81Ø80.924`, 0.383`<, 2Ø80.383`, 0.924`<, 3Ø8-0.383`, 0.924`<, 4Ø8-0.924`, 0.383`<, 5Ø8-0.924`,-0.383`<, 6Ø8-0.383`,-0.924`<, 7Ø80.383`,-0.924`<, 8Ø80.924`,-0.383`<, 9Ø81.848`, 0.765`<, 10Ø80.765`, 1.848`<, 11Ø8-0.765`, 1.848`<, 12Ø8-1.848`, 0.765`<, 13Ø8-1.848`,-0.765`<, 14Ø8-0.765`,-1.848`<, 15Ø80.765`,-1.848`<, 16Ø81.848`,-0.765`<<<;

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881Ø2, 1Ø9, 1Ø16, 2Ø3, 2Ø5, 3Ø7, 3Ø18, 4Ø5, 4Ø10, 4Ø13, 5Ø6, 6Ø12, 6Ø15, 7Ø8, 7Ø10, 8Ø9, 8Ø11, 9Ø12, 10Ø14, 11Ø13, 11Ø15, 12Ø14, 13Ø16, 14Ø17, 15Ø18, 16Ø17, 17Ø18<,

81Ø80, 0<, 2Ø80, 3<, 3Ø80, 6<, 4Ø81, 1<, 5Ø81, 3<, 6Ø81, 5<, 7Ø82, 2<, 8Ø82, 3<, 9Ø82, 4<, 10Ø83, 2<, 11Ø83, 3<, 12Ø83, 4<, 13Ø85, 1<, 14Ø84, 3<,

15Ø85, 5<, 16Ø86, 0<, 17Ø86, 3<, 18Ø86, 6<<<;

881Ø2, 1Ø5, 1Ø10, 2Ø7, 2Ø9, 3Ø4, 3Ø6, 3Ø11, 4Ø5, 4Ø8, 5Ø14, 6Ø7, 6Ø12, 7Ø16, 8Ø9, 8Ø13, 9Ø18, 10Ø11, 10Ø17, 11Ø15, 12Ø13, 12Ø17, 13Ø14, 14Ø16, 15Ø16, 15Ø18, 17Ø18<,

81Ø80, 0<, 2Ø80, 6<, 3Ø81, 1<, 4Ø81, 3<, 5Ø81, 4<, 6Ø82, 2<, 7Ø82, 5<, 8Ø83, 3<, 9Ø83, 6<, 10Ø84, 0<, 11Ø84, 1<, 12Ø84, 2<, 13Ø84, 3<, 14Ø84, 4<,

15Ø85, 2<, 16Ø85, 5<, 17Ø86, 0<, 18Ø86, 6<<<;

881Ø2, 1Ø6, 1Ø20, 2Ø3, 2Ø17, 3Ø4, 3Ø12, 4Ø5, 4Ø15, 5Ø6, 5Ø10, 6Ø7, 7Ø8, 7Ø16, 8Ø9, 8Ø19, 9Ø10, 9Ø14, 10Ø11, 11Ø12, 11Ø20, 12Ø13, 13Ø14, 13Ø18, 14Ø15, 15Ø16, 16Ø17, 17Ø18, 18Ø19, 19Ø20<, Thread@

Range@20DØ 880, 0<,80, 4<,80, 6<,81, 5<,81, 2<,81, 1<,

82, 1<,84, 1<,84, 3<,83, 2<,86, 2<,86, 6<,85, 5<,

84, 5<,82, 5<,82, 3<,83, 4<,85, 4<,85, 1<,86, 0<<D<;

881Ø2, 1Ø9, 1Ø18, 2Ø5, 2Ø8, 3Ø4, 3Ø9, 3Ø11, 4Ø5, 4Ø6, 5Ø13, 6Ø7, 6Ø16, 7Ø8, 7Ø10, 8Ø20, 9Ø10, 10Ø14, 11Ø12, 11Ø15, 12Ø13, 12Ø14, 13Ø17, 14Ø19, 15Ø16, 15Ø18, 16Ø17, 17Ø20, 18Ø19, 19Ø20<,

81Ø80, 0<, 2Ø80, 7<, 3Ø81, 2<, 4Ø81, 3<, 5Ø81, 6<, 6Ø82, 3<, 7Ø82, 4<, 8Ø82, 7<, 9Ø83, 1<, 10Ø83, 4<, 11Ø84, 2<, 12Ø84, 5<, 13Ø84, 6<, 14Ø85, 4<, 15Ø86, 2<, 16Ø86, 3<, 17Ø86, 6<, 18Ø87, 0<, 19Ø87, 4<, 20Ø87, 7<<<;

881Ø2, 1Ø3, 1Ø18, 2Ø6, 2Ø12, 3Ø4, 3Ø13, 4Ø5, 4Ø10, 5Ø6, 5Ø8, 6Ø20, 7Ø8, 7Ø13, 7Ø16, 8Ø9, 9Ø11, 9Ø12, 10Ø11, 10Ø19, 11Ø14, 12Ø15, 13Ø14, 14Ø17, 15Ø16, 15Ø17, 16Ø18, 17Ø20, 18Ø19, 19Ø20<,

81Ø80, 0<, 2Ø80, 6<, 3Ø81, 1<, 4Ø81, 3<, 5Ø81, 5<, 6Ø81, 6<, 7Ø82, 2<, 8Ø82, 5<, 9Ø83, 5<, 10Ø83, 3<, 11Ø83, 4<, 12Ø84, 5<, 13Ø84, 1<, 14Ø84, 4<, 15Ø85, 4<, 16Ø85, 2<, 17Ø85, 5<, 18Ø86, 0<, 19Ø86, 3<, 20Ø86, 6<<<;

(15)

881Ø4, 1Ø15, 1Ø21, 2Ø3, 2Ø8, 2Ø22, 3Ø5, 3Ø6, 4Ø7, 4Ø8, 5Ø9, 5Ø21, 6Ø7, 6Ø10, 7Ø17, 8Ø11, 9Ø12, 9Ø13, 10Ø12, 10Ø14, 11Ø13, 11Ø14, 12Ø15, 13Ø16, 14Ø18, 15Ø19, 16Ø17, 16Ø19, 17Ø20, 18Ø20, 18Ø22, 19Ø22, 20Ø21<,

81Ø80, 0<, 2Ø80, 7<, 3Ø81, 2<, 4Ø81, 5<, 5Ø82, 1<, 6Ø82, 3<, 7Ø82, 4<, 8Ø82, 6<, 9Ø83, 2<, 10Ø83, 3<, 11Ø83, 5<, 12Ø84, 2<, 13Ø84, 3<, 14Ø84, 5<,

15Ø85, 1<, 16Ø85, 3<, 17Ø85, 4<, 18Ø85, 6<, 19Ø86, 2<, 20Ø86, 5<, 21Ø87, 0<, 22Ø87, 7<<<;

881Ø2, 2Ø3, 3Ø4, 4Ø5, 5Ø6, 6Ø7, 7Ø8, 8Ø9, 9Ø10, 10Ø11, 11Ø12, 12Ø13, 13Ø14, 14Ø15, 15Ø16, 16Ø17, 17Ø18, 18Ø19, 19Ø20, 20Ø21, 21Ø22, 22Ø1, 1Ø15, 2Ø8, 3Ø17, 4Ø21, 5Ø11, 6Ø18, 7Ø13, 9Ø20, 10Ø16, 12Ø22, 14Ø19<, Thread@Range@22DØ 880, 0<,82, 0<,87, 0<,87, 7<,

86, 6<,86, 5<,82, 5<,82, 3<,84, 3<,85, 3<,

85, 6<,81, 6<,81, 5<,81, 2<,81, 1<,85, 1<,

86, 1<,86, 2<,83, 2<,83, 4<,83, 7<,80, 7<<D<;

881Ø7, 1Ø13, 1Ø21, 2Ø4, 2Ø19, 2Ø22, 3Ø4, 3Ø8, 3Ø10, 4Ø5, 5Ø6, 5Ø15, 6Ø7, 6Ø9, 7Ø12, 8Ø9, 8Ø13, 9Ø18, 10Ø11, 10Ø14, 11Ø12, 11Ø17, 12Ø19, 13Ø20, 14Ø15, 14Ø21, 15Ø16, 16Ø17, 16Ø20, 17Ø18, 18Ø22, 19Ø20, 21Ø22<,

Thread@Range@22DØ880, 0<,88, 0<,81, 2<,82, 2<,

83, 2<,86, 2<,87, 2<,81, 3<,86, 3<,82, 4<,

85, 4<,87, 4<,81, 6<,82, 6<,83, 6<,84, 6<,

85, 6<,86, 6<,87, 6<,84, 7<,80, 8<,88, 8<<D<;

881Ø2, 1Ø5, 1Ø18, 2Ø7, 2Ø22, 3Ø4, 3Ø8, 3Ø21, 4Ø5, 4Ø16, 5Ø9, 6Ø7, 6Ø8, 6Ø10, 7Ø17, 8Ø11, 9Ø10, 9Ø12, 10Ø20, 11Ø12, 11Ø14, 12Ø13, 13Ø17, 13Ø19, 14Ø15, 14Ø18, 15Ø16, 15Ø20, 16Ø17, 18Ø19, 19Ø21, 20Ø22, 21Ø22<,

81Ø80, 0<, 2Ø80, 6<, 3Ø81, 1<, 4Ø81, 2<, 5Ø81, 3<, 6Ø81, 4<, 7Ø81, 5<, 8Ø82, 1<, 9Ø82, 3<, 10Ø82, 4<, 11Ø83, 1<, 12Ø83, 3<, 13Ø83, 4<, 14Ø84, 1<,

15Ø84, 2<, 16Ø84, 3<, 17Ø84, 4<, 18Ø85, 1<, 19Ø85, 3<, 20Ø85, 5<, 21Ø86, 0<, 22Ø86, 6<<<;

(16)

881Ø4, 1Ø7, 1Ø23, 2Ø3, 2Ø12, 2Ø24, 3Ø5, 3Ø8, 4Ø6, 4Ø12, 5Ø7, 5Ø10, 6Ø9, 6Ø11, 7Ø14, 8Ø9, 8Ø13, 9Ø15, 10Ø11, 10Ø16, 11Ø18, 12Ø17, 13Ø19, 13Ø23, 14Ø15, 14Ø21, 15Ø20, 16Ø17, 16Ø19, 17Ø20, 18Ø22, 18Ø24, 19Ø21, 20Ø22, 21Ø24, 22Ø23<,

81Ø80, 0<, 2Ø80, 7<, 3Ø81, 3<, 4Ø81, 4<, 5Ø82, 3<, 6Ø82, 4<, 7Ø83, 1<, 8Ø83, 2<, 9Ø83, 3<, 10Ø83, 4<, 11Ø83, 5<, 12Ø83, 6<, 13Ø84, 1<, 14Ø84, 2<, 15Ø84, 3<, 16Ø84, 4<, 17Ø84, 5<, 18Ø84, 6<, 19Ø85, 3<, 20Ø85, 4<, 21Ø86, 3<, 22Ø86, 4<, 23Ø87, 0<, 24Ø87, 7<<<;

881Ø2, 1Ø3, 1Ø4, 2Ø5, 2Ø6, 3Ø19, 3Ø23, 5Ø7, 5Ø8, 6Ø18, 7Ø12, 9Ø7, 9Ø10, 9Ø11, 10Ø13, 10Ø4, 11Ø17, 11Ø19, 12Ø22, 13Ø14, 13Ø15, 14Ø16, 14Ø8, 15Ø6, 15Ø17, 16Ø24, 17Ø20, 18Ø12, 18Ø21, 19Ø21, 20Ø22, 21Ø16, 22Ø24, 23Ø20, 23Ø8, 24Ø4<, Thread@Range@24DØ880, 0<,80, 4<,85, 1<,87, 0<,

80, 7<,83, 4<,81, 5<,87, 7<,81, 2<,82, 1<,84, 2<,

84, 5<,82, 3<,82, 6<,83, 3<,85, 6<,84, 3<,84, 4<,

85, 2<,86, 3<,85, 4<,86, 5<,86, 1<,86, 6<<D<;

(17)

CrossingNumberGraph8E=

CrossingNumberGraph10A =

881Ø5, 1Ø7, 1Ø25, 2Ø3, 2Ø6, 2Ø24, 3Ø4, 3Ø12, 4Ø8, 4Ø11, 5Ø6, 5Ø9, 6Ø15, 7Ø8, 7Ø22, 8Ø14, 9Ø10, 9Ø13, 10Ø11, 10Ø18, 11Ø21, 12Ø13, 12Ø16, 13Ø19, 14Ø15, 14Ø17, 15Ø20, 16Ø17, 16Ø25, 17Ø18, 18Ø23, 19Ø20, 19Ø22, 20Ø21, 21Ø26, 22Ø23, 23Ø24, 24Ø26, 25Ø26<,

81Ø80, 0<, 2Ø80, 8<, 3Ø82, 2<, 4Ø82, 3<, 5Ø82, 4<, 6Ø82, 7<, 7Ø84, 1<, 8Ø84, 3<, 9Ø84, 4<, 10Ø84, 5<, 11Ø84, 6<, 12Ø86, 2<, 13Ø86, 3<, 14Ø86, 4<,

15Ø86, 6<, 16Ø88, 2<, 17Ø88, 4<, 18Ø88, 5<, 19Ø810, 3<, 20Ø810, 6<, 21Ø810, 7<, 22Ø812, 2<, 23Ø812, 5<, 24Ø812, 8<, 25Ø814, 0<, 26Ø814, 7<<<;

881Ø2, 1Ø8, 1Ø24, 2Ø5, 2Ø28, 3Ø6, 3Ø7, 3Ø27, 4Ø5, 4Ø7, 4Ø10, 5Ø18, 6Ø12, 6Ø19, 7Ø11, 8Ø9, 8Ø14, 9Ø10, 9Ø12, 10Ø23, 11Ø15, 11Ø20, 12Ø13, 13Ø16, 13Ø26, 14Ø15, 14Ø21, 15Ø16, 16Ø18,

17Ø18, 17Ø19, 17Ø22, 19Ø24, 20Ø24, 20Ø25, 21Ø22, 21Ø27, 22Ø23, 23Ø25, 25Ø26, 26Ø28, 27Ø28<,

81Ø80, 0<, 2Ø80, 8<, 3Ø81, 1<, 4Ø81, 4<, 5Ø81, 7<, 6Ø82, 1<, 7Ø82, 2<, 8Ø82, 3<, 9Ø82, 4<, 10Ø82, 6<, 11Ø83, 2<, 12Ø83, 4<, 13Ø83, 5<, 14Ø84, 3<, 15Ø84, 4<, 16Ø84, 5<, 17Ø85, 4<, 18Ø85, 5<, 19Ø86, 1<, 20Ø86, 2<, 21Ø86, 3<, 22Ø86, 4<, 23Ø86, 6<, 24Ø87, 1<,

25Ø87, 4<, 26Ø87, 7<, 27Ø88, 0<, 28Ø88, 8<<<;

(18)

881Ø2, 1Ø6, 1Ø29, 2Ø4, 2Ø26, 3Ø4, 3Ø5, 3Ø9, 4Ø17, 5Ø12, 5Ø13, 6Ø7, 6Ø11, 7Ø8, 7Ø12, 8Ø24, 8Ø28, 9Ø10, 9Ø27, 10Ø11, 10Ø18, 11Ø15, 12Ø20, 13Ø14, 13Ø22, 14Ø15, 14Ø25, 15Ø16, 16Ø17, 16Ø21, 17Ø24, 18Ø19, 18Ø23, 19Ø20, 19Ø26, 20Ø21, 21Ø30, 22Ø23, 22Ø29, 23Ø24, 25Ø26, 25Ø28, 27Ø28, 27Ø30, 29Ø30<,

81Ø80, 0<, 2Ø80, 9<, 3Ø81, 1<, 4Ø81, 7<, 5Ø82, 2<, 6Ø82, 5<, 7Ø82, 6<, 8Ø82, 8<, 9Ø83, 1<, 10Ø83, 4<, 11Ø83, 5<, 12Ø83, 6<, 13Ø84, 2<, 14Ø84, 3<, 15Ø84, 5<, 16Ø84, 6<, 17Ø84, 7<, 18Ø85, 3<, 19Ø85, 4<, 20Ø85, 5<, 21Ø85, 6<, 22Ø86, 2<, 23Ø86, 3<, 24Ø86, 7<, 25Ø87, 2<, 26Ø87, 7<, 27Ø88, 1<, 28Ø88, 8<, 29Ø89, 0<, 30Ø89, 9<<<;