1. Ibrahim CahitIbrahim Cahit
Near 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. 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
17. 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. 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. 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. 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. 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.
23. 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. 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. 25
Antimatroids: Definition
• Let A be a set, L a set of strings over A
• (A,L) is an antimatroid if
1. (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. 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. 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. 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. 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. 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.
a
L = {a,
ab,
abc,
abcd,
abcde,
b
c
d
e
f
...}
abcdef,
31. 31
Haken and Appel needed a computer in 1976….
http://www.mathpuzzle.com
32. 32
Bad example No.1
(Heawood graph, 1898)
2
4 2 4
2 1
4
1
3
2
3
3
1
4 3
2
4
2
4
1
3
4 1
3
1
Red-Yellow
chain
Green-Blue
chain
Hamilton cycle in the
dual graph (closest
triangle first)
By using Hamiltonian cycle.
33. 33
Bad example
(Heawood graph)
2
4 3
2
3
1
2
1
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. 34
Bad example No. 2
(Errera graph, 1921)
4
1
3
2
2
3
1
4 3
2
3
2
4
1
4
1
3
Spiral
chain:
Hamilton
path in the
dual graph
37. 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. 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.
40. 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. 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
45. 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 81012141618
19
20
21
3433323130
Coloring the fan F3
by red-green chain
Coloring the fan F1 by
yellow-green-blue chain
46. 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
3
Gi-1
Cycle at the i th level
4 3 1
Coloring outercycle with the spiral chain
(termination condition)
47. 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
51. 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. 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 2
Spiral chain 4
Spiral chain 3
Spiral chain 5
This case is handled mainly by 3-reducibility (A. Parmar)
53. 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. 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. 55
Evolution to a three coloring
(A Stochastic Cellular Automaton for Three Coloring Penrose Tilings,
Mark McClure, 2001)
56. 56
Spiral Chains and Three Coloring Penrose Tilings
: Roger Penrose (1973)
With kites and darts With rhombs
58. 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. 59
Example (Four Great Circles)
A
A'
B
B'
C
C'
D
E
F
G
H
I
C'
I D
A'
B
A C
H
E
G F
B'
C2 C1
C3
C4
60. 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
62. 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. 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.
65. 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
70. 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. 71
Some known results
• (Robertson, Seymour, Thomas) Every
minimal counterexample to Hadwiger’s
conjecture for k=5 is apex (Gv 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. 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. 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”?
77. 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. 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. 79
Spiral Chains:
A New Proof of the Four Color Theorem
Ibrahim Cahit
Near East University
Thank you
Istanbul Bilgi University, 13 May 2005
