Ibrahim CahitIbrahim CahitNear East UniversityNear East University

Spiral Chains: Spiral Chains: The Four Color Theorem and BeyondThe Four Color Theorem and Beyond

Istanbul Bilgi University, 13 May 2005

2

Presentation Plan�

Problem definition

�

Some map coloring examples�

Inspiration and motivation (Why spiral chains?)

�

Historical notes and Kempe’s

idea�

Use of spiral chains in graphs

�

Proof outline�

Three-Coloring Penrose Tilings

�

Coloring of Arrangements of Great Circles�

Steinberg’s Three

Coloring

Conjecture

�

Hadwiger’s Conjecture�

Tribute and concluding remarks

3

Proper Coloring of the Maps

Red

Yellow

Green

Blue

4

Proper Coloring of the Maps

Red

Yellow

Green

Blue

5

Non-trivial 4-coloring!

?There is a better 4-coloring of USA map than this one!

6

Another USA Map Coloring

There is a better 4-coloring of USA map than this one!

7

Still another USA map 4-coloring!

8

A difficult one

No two colors can be exchanged!

9

You may work on the actual map(Martin Gardner’s the April Fool’s hoax map, 1975)

or…Courtesy of Juan Orozco, Boston 2003.

10

Non-proper map coloring with 4-colors!

Not all coloring algorithms terminate with a proper coloring.

11

The Spiral in Nature … (The Geometry Junkyard:Spirals) http://www.ics.uci.edu/~eppstein/junkyard/spiral.html

1 mm

12

Spirals in the geometry

…

(The Geometry Junkyard) http://www.ics.uci.edu/~eppstein/junkyard/spiral.html

13

Spirals in the Arts …

(The Geometry Junkyard:Spirals) http://www.ics.uci.edu/~eppstein/junkyard/spiral.html

14

Spirals in the Universe… (Spiral Galaxies)

15

Spiral Chains and the Four Color Theorem

16

3

4

3

1

2

2

11 2

1

2

13

4

43

4

2

3

2

34

231

21

4

3

1

1

43 4

2

2

v1

v7

v22

v36

Spiral segment 1(full-revolution),{Yellow, Red, Blue}

Spiral segment 2 (full-revolution), {Green, Yellow, Red}

Spiral segment 3,{Red, Blue, Yellow}

Spirals in graph theory

I. Cahit (2003)

Historical notes next …

17

The Four Color Problem has a long story

�

Francis Guthrie

(1852) (problem owner)

�

A. de Morgan (first mathematical look)

�

Arthur Cayley (1878) (first paper)

�

Alfred Kempe (1879) (first proof)

�

P. Heawood (1890) (refuted Kempe’s “proof”)

�

P. G. Tait (1880) (another “proof”)

�

Petersen (1891) (refuted Tait’s “proof”)

�

G. Birkhoff (1913) (first reducible configuration)

18

The Four Color Problem has a long story (2)

Reducibility�

Franklin, Bernhard and Bernhard, Reylonds, Winn, Ore and Stample, Ore, Stromquist, Meyer, Tutte, Whitney, Allaire, Swart, Düre, Heesch, Miehe

�

Henrich Heesch, Jean Meyer

Discharging

19

The Four Color Problem has a long story (3) Proofs at last …

�

Kenneth Appel and Wolfgang Haken (1976)

�

N. Robertson, D. Sanders, P. Seymour, and R. Thomas (1996)

� Proofs use computers

� Proofs are complicated

� Skepticism

20

Who is responsible for this extremely lengthy and computer assisted proofs ?

�

George Birkhoff (1913)!

�

Why?(C-reducibility in A&H proof)

The Birkhoff Diamond

Ring

21

Kempe’s

Idea

�

Use of Kempe chain to color white face.

�

Coloring maps and the Kowalski doctrine (John McCarthy, 1982).

�

Use of shelling structures (antimatroids) in map coloring (A. Parmar, 2003).

�

Our coloring algorithm is an antimatroid without backtracking.

22

Kempe’s

argument

Red-Green ChainRed-Yellow Chain

23

Philosophy of the known approach…

�

A configuration is reducible if it cannot be contained in a minimum counterexample to the four-color conjecture.

�

The proof by A&H is actually set up as a contraposition of the inductive step;

the

“minimal”

counterexample is the smallest graph for which the inductive step cannot be made. [D. Pavlovic]

Shelling structures next…

24

Greedoids* (http://www.formal.stanford.edu/aarati)

�

Greedoids : Mathematical structures under which greedy algorithms reach optimal solutions

�

Two kinds:

–

Matroids: structure underlying greedy algorithms for finding minimum spanning tree of graph

–

Antimatroids: (shelling structures), can be decomposed by removing successive layers until nothing is left

*Aarati Parmar, *Aarati Parmar, ““Some Mathematical Structures Underlying Efficient PlanningSome Mathematical Structures Underlying Efficient Planning””, Stanford , Stanford University, March 2003.University, March 2003.

25

Antimatroids: Definition

�

Let A be a set, L a set of strings over A

�

(A,L) is an antimatroid if1. (Simple) No string in L has a repeated element of A

2. (Normal) Every symbol of A appears in some word of L

3. (Hereditary) L is closed under prefixes

4. (Exchange) If s, t are words of L, and s contains an element of A not in t, then for some x in s-t, tx is a word of L

26

Four-Coloring Maps

�

Antimatroids (shelling structures) can be decomposed by removing successive layers until nothing is left.

�

Antimatroid structure shows us when we can effect planning without search!

�

Heuristics

of postponing coloring show us how to order subgoals in such a way as to avoid any dependencies.

27

Four-Coloring Maps

�

A graph (V,E) is n-reducible if one can repeatedly remove vertices of degree n

or less, resulting in the

empty graph.

�

If a graph is n-reducible then we can color it with n+1 colors without backtracking.

�

Let L(V,E)

be the shelling sequences of removing the vertices of degree n

or less from (V,E)

�

Theorem : (V,E) n-reducible iff L(V,E)

is an antimatroid.

28

Four-Coloring Maps, con’t.

�

Strategy: postpone 4-coloring countries with 3 or fewer neighbors; remove from map; repeat

�

If entire map is decomposed in this way, the reverse order is a plan for coloring the map!

�

“Color California last.”

�

When do maps have this property?

29

Four-Coloring Maps

�

We want to know when we can color without having to backtrack

�

Idea in [Kempe, 1879], [McCarthy 1982]:1. postpone 4-coloring countries with 3 or fewer

neighbors;

2. remove from map;

3. repeat.

30

Antimatroids = Shelling sequences

�

If L is simple and normal, equivalent to shellings of convex geometries in Euclidean spaces

�

In our algorithm L is union of sub-spiral chains of a MPG.

aL = {a,

ab,abc,abcd,abcde,

b

cd

ef

...}

abcdef,

31

Haken and Appel needed a computer in 1976….http://www.mathpuzzle.com

32

Bad example No.1 (Heawood graph, 1898)

2

4 2 4

2 1

4

1

3

23

3

1

4 3

2

4

2

4

1

3

4 1

3

1

Red-Yellowchain

Green-Bluechain

Hamilton cycle in thedual graph (closest

triangle first)

By using Hamiltonian cycle.

33

Bad example (Heawood graph)

2

4 3

2

3

1

21

3

2

4 1

3

4

2

4

1

2

4 3

1

3

1

2

4

Spiral chain 1

Spiral chain 2

By using spiral chains

Theta sub-graph separates two spiral chains

34

Bad example No. 2 (Errera graph, 1921)

4

1

32

23

1

4 3

2

3

2

4

1

4

1

3

Spiralchain:

Hamiltonpath in thedual graph

35

Bad example No.3 (Kittell graph, 1935)

36

Spiral Chain=Shelling StructureS={ 2, 1, 2, …

{2,1,2,3,4,3,4,3,4,3,…

{2,1,2,3,4,3,4,3,4,3,1,2,1,2,1,…

{2,1,2,3,4,3,4,3,4,3,1,2,1,2,1,4,3}

4

1

32

23

1

4 3

2

3

2

4

1

4

1

3

37

Algorithmic proof based on the spiral chains

�

Theorem. All maximal planar graphs are 4-colorable by the use of spiral chains.

Proof:

Case (a) Maximal planar graphs with a single spiral chain.

Case (b) Maximal planar graphs with several spiral chains.

38

A node on the spiral chain

�

It

looks

like

a

way

to

"cut

up" a graph so that

each

node

is connected

to

one

of four

kinds

of

nodes: one

node

forward

of it in the

spiral, and

one

node

behind

it, and

then

a

set of nodes

to

its

"right" which

are bisected

by

the

chain, and

another

set on

the

"left" which

are

bisected

on the

other side.

39

Coloring a node

X

4

2

1

3

2

4

3

2

Si-1

Si

Si+1

Right sub-spiralchain

Left sub-spiralchain

40

Close look at the spiral chain …

Direction of coloring of the nodes on the spiral chain*

Start node

End node

* Along with the spiral chain use whenever possible 2 (possible) colors e.g., blue-yellow, green-red, etc. Otherwise use 3 colors.

41

Proof Without Words

Spiral Segment 1

Spiral Segment 2

Spiral Segment 3

STEP 1:Spiral Chain of the Maximal Planar Graph

STEP 2:THREE COLORING OF SPIRAL SEGMENTS

MAXIMAL OUTERPLANAR

SUB-GRAPH

42

The use of “safe”

colors

43

Theta-Separator Sub-graph

1

2

yx

z

t

(b)

S1

S2

-separator subgraphθ

x

y

(a)

Triangulated outer-planar graphs

44

Fan decomposition

1

i-typefan with2 faces

o-typefan with3 faces

2

3

4

56 7

8

9

10111213

45

Ordering the triangles in the fans for 4-coloring

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1 3 5 7 9 11 13 15 17

42 6 8 1012141618

19

20

21

3433323130

Coloring the fan F3

by red-green chain

Coloring the fan F1

by yellow-green-blue chain

46

Extending 4-coloring to the outer-cycle

�

i th level cycle nodes must be colored

with

at

most three colors e.g., green , red, yellow in order to reserve a color for the last node of the spiral chain (shown in blue )

1 3

2

4

3

4

3Gi-1

Cycle at the i th level

4 3 1

Coloring outercycle with the spiral chain(termination condition)

47

Spiral chains and fan decompostion in a Zoe graph

1

2

3

4

5 6 7

8

9

10

11

12

1314

15

16

17

18

19

20

21

48

Yuchun Li (China)

programmed the spirals, (1.11.2004)

49

Primary Colors: Red,Blue ,Green“SAFE”

Color : Yellow

OUTER NODE

SPIRAL CHAIN “THREE-PLUS-ONE” COLORING

Another spiral chain coloring algorithm

Some Other Coloring Problems with the Spiral Chains

51

Uniquely 4-colorable graphs

�

Start with K4 and add a new node joined to 3 nodes of a face;

repeat.

�

Theorem (Fowler, conjectured by Fisk and Fiorini-Wilson 1977)

Every uniquely 4- colorable planar graph can be obtained as described above.

52

Spiral chains in an uniquely 4-colorable MPG

1(1)

4(3) 4(3)2(3) 3(3)

3(3)

4(3)

1(2)

1(3)

3(2)

4(1) 1(3)

2(2)

2(3)

3(1)2(1)

Spiral chain 1

Spiral chain 2Spiral chain 4

Spiral chain 3

Spiral chain 5

This case is handled mainly by 3-reducibility (A. Parmar)

53

Spiral chains and 3-coloring of Tutte’s graph, 1954 (Counter-example to Tait’s conjecture, 1880)

Spiral chain 1

Spiral

chain 2

Spiral chain 3

Tait Conjecture:

Every cubic graph is Hamiltonian.

54

Coloring Penrose Tilings

�

Three coloring of Penrose tiles proposed by J. H. Conway.

�

Simpler than the four color problem.

�

Regions are in the form of kite and dart, rhombs or pentacles only.

�

Open problem whether Penrose pentacles tiles are 3- colorable.

τ

τ τ

τ

1

1

1

1

The Kite The Dart

55

Evolution to a three coloring (A Stochastic Cellular Automaton for Three Coloring Penrose Tilings,

Mark McClure, 2001)

56

Spiral Chains and Three Coloring Penrose Tilings: Roger Penrose (1973)

With kites and darts With rhombs

57

Three colored tiling by pentacles

58

3-Colorability of Arrangements of Great Circles (Stan Wagon, 2000)

�

Is every

zonohedron

face 3-colorable

when

viewed

as a planar

map? An equivalent

question,

under

a different

guise, is the

following: is the

arrangement

graph of great

circles

on the

sphere

always

vertex

3- colorable? YES (next slide)

�

Can spiral chains be any help?

YES (next slide)

A

A'

B

B'

C

C'

D

E

F

F'

E'

D'1b

1f

3f

5f

5b

4b

3b

2f

4f

2b

Triangular chain # 1= {(1b &1f), (1f & 2b), (2b & 4b), (4b & 3b), (3b &1b)}Triangular chain # 2= {(5b & 5f), (5f & 3f), (3f & 4f), (4f & 2f), (2f & 5b)}

59

Example (Four Great Circles)

A

A'

B

B'

C

C'

D

E

F

G

H

I

C'

I D

A'

B

A CH

E

G F

B'

C2 C1

C3C4

60

Three Coloring of Arrangements of Great Circles by Spiral Chains

1

23

4

5 6

78

9

Decomposition into triangles and 3-coloring

Note:

(Number of triangles) /2 = Number of great circles

61

Four Coloring of the Koesten

Graph Using Spiral Chains

62

Steinberg’s Conjecture

(1973)

�

(Steinberg) Every planar graph without cycles of length 4 and 5 is 3-colorable.

�

(Borodin et. al.2005)

Every planar graph without cycles of length 4 to 7 is 3- colorable.

63

Proof attempt

�

Characterization of planar graphs with cycles 4 and 5 that are not 3-colorable.

�

Extending these graphs to 3-colorable graphs by deleting suitable edges.

�

Use of spiral-chain coloring to show that planar graphs without 4 and 5 cycles are 3-colorable.

64

Edge-driven coloring

3C

3C

5C 4C

4C

4C

(a) (b)

(c) (d)

(e) (f)

|Cout|=4

|Cout|=6

|Cin|+|Cout|=12

|Cout|=7

{o,o,o,o,e,o,e,o}

|Cin|+|Cout|=9 {o,o,o,o,e,o,o,o}

4C to 3C by edge deletion

|Cin|+|Cout|=10

65

An triangulated ring is 3-colorable only if

| | 0 or | | | | 0(mod3)o o iC C C≡ + ≡Cyclic parity sequence of the fans around the inner cycle Ci

is symmetric and

66

1 2

C(out)

|C(out)|+|C1|+|C2|=0 (mod3)

P(C1)={e,o,o,o,o,o,o,o,o,o,o,e}

P(C2)={e,o,o,o,o,o,o,e}

67

Spiral-chain coloring around triangle, square and pentagon

68

Spiral Chain

Coloring

Without cycle size in {4,5,…,9}

69

Spiral Chains: Future Work

70

Hadwiger’s Conjecture

�

Hadwiger

(1943): Graphs containing no Kk+1

-minor are k-colorable.

�

Trivial for k<4.

�

Equivalent to 4CC for k=4 (Wagner, 1937) and for k=5 (Robertson, Seymour and Thomas, 1993).

�

Open for k>5.

71

Some known results

�

(Robertson, Seymour, Thomas)

Every minimal counterexample to Hadwiger’s

conjecture for k=5 is apex (G\v is planar for some vertex v of G).

�

Hajos’

Conjecture. Every loopless

graph with no Kk+1 subdivison

is k-colorable.

(k≤

3 ≡

Hadwiger

(true))(k= 4,5,6 (Open))(k≥

7 false (Catlin)).

72

Embedding

Kn

and

Spiral Chains

K5 K6

Embed K6

-free graph G in the plane so that edges of every K4

remain in between two spiral segments.

73

Back to 4CT… Georges Gonthier, “A computer-checked proof of the Four

Colour

Theorem”, 2005.

�

… fully checked by the Coq v7.3.1 proof assistant. This proof is largely based on the mixed mathematics/computer proof of Robertson et. al. but contains original parts.

�

57p+28p+?=???�

For someone this is the end of the “skepticism”

e.g.,

Devlin.

�

Now, is it “humanly readable”?

74

Tribute

Francis Guthrie A. de Morgan

75

Tribute

Kenneth Appel Wolfgang Haken

76

Tribute

77

Concluding Remarks�

Spiral chains in graphs introduced.

�

Non-computer proof of the 4CT has been given. (Ideas can only be created by humans).

�

The use of the spiral chains to the other graph coloring problems demonstrated.

78

Acknowledgements�

Louis Kaufmann

(G. Washington Univ.)

�

Aarati Parmar

(Stanford University)

�

Juan Orozco

(Boston)

�

Stan Wagon

(McCalester

College)

�

Mehmet Özel (Lefkose)

�

Shel Hulac (Girne American Univ.)

�

Chris Heckman (Arizona State)

For

their support and comments on spiral chains …

79

Spiral Chains: A New Proof of the Four Color Theorem

Ibrahim Cahit

Near East University

Thank you

Istanbul Bilgi University, 13 May 2005