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The Seven Colour Theorem
Christopher Tuffley
Institute of Fundamental SciencesMassey University, Palmerston North
3rd Annual NZMASP Conference, November 2008
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 1 / 17
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Outline
1 Introduction
Map colouring
2 The torus
From maps to graphsEuler characteristic
Average degree
Necessity and sufficiency
3 Other surfacesRevisiting the plane
The Heawood bound
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Introduction Map colouring
Map colouring
How many crayons do you need to colour Australia. . .
. . . if adjacent regions must be different colours?
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Introduction Map colouring
Map colouring
How many crayons do you need to colour Australia. . .
. . . if adjacent regions must be different colours?
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 3 / 17
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Introduction Map colouring
Map colouring
How many crayons do you need to colour Australia. . .
. . . if adjacent regions must be different colours?
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 3 / 17
I d i M l i
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Introduction Map colouring
Four colors suffice
Theorem (Appel and Haken, 1976)
Four colours are necessary and sufficient to properly colour
maps drawn in the plane.
Some maps require four colours (easy!)
No map requires more than four colours (hard!).
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 4 / 17
I t d ti M l i
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Introduction Map colouring
Four colors suffice
Theorem (Appel and Haken, 1976)
Four colours arenecessaryand sufficient to properly colour
maps drawn in the plane.
Some maps require four colours (easy!)
No map requires more than four colours (hard!).
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 4 / 17
Introduction Map colouring
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Introduction Map colouring
Four colors suffice
Theorem (Appel and Haken, 1976)
Four colours are necessary andsufficientto properly colour
maps drawn in the plane.
Some maps require four colours (easy!)
No map requires more than four colours (hard!).
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 4 / 17
Introduction Map colouring
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Introduction Map colouring
On the donut they do nut!
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17
Introduction Map colouring
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Introduction Map colouring
On the donut they do nut!
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17
Introduction Map colouring
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Introduction Map colouring
On the donut they do nut!
How many colours do we need??
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17
The torus
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The torus
The Seven Colour Theorem
Theorem
Seven colours are necessary and sufficient
to properly colour maps on a torus.
Steps:
1 Simplify!
2 Use the Euler characteristicto find the average degree.
3 Look at a minimal counterexample.
4 Prove necessity.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 6 / 17
The torus From maps to graphs
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p g p
From maps to graphs
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17
The torus From maps to graphs
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p g p
From maps to graphs
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17
The torus From maps to graphs
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From maps to graphs
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17
The torus From maps to graphs
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From maps to graphs
The dual of the map
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17
The torus Euler characteristic
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Euler characteristic
S a surfaceG a graph drawn on S so that
no edges or vertices crossor overlapall regions (faces) are discs
there areV vertices, E edges, F faces.
Definition
The Euler characteristicof S is (S) = V E+ F.
Theorem
(S) depends only on S and not on G.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 8 / 17
The torus Euler characteristic
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Euler characteristic
S a surfaceG a graph drawn on S so that
no edges or vertices crossor overlapall regions (faces) are discs
there areV vertices, E edges, F faces. 00000000
00000000
00000000
00000000
00000000
00000000
11111111
11111111
11111111
11111111
11111111
11111111
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
11111111
11111111
11111111
11111111
11111111
11111111
11111111
11111111
Definition
The Euler characteristicof S is (S) = V E+ F.
Theorem
(S) depends only on S and not on G.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 8 / 17
The torus Euler characteristic
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Examples
(torus) = 1 2 + 1 = 0 (sphere) = 4 6 + 4 = 2
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The torus Euler characteristic
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Proof of invariance
Given graphs G1 and G2, find a common refinement H.
Subdivide edges
Add vertices in facesSubdivide faces.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17
The torus Euler characteristic
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Proof of invariance
Given graphs G1 and G2, find a common refinement H.
Subdivide edges
Add vertices in facesSubdivide faces.
V E F
1 1 0 0
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The torus Euler characteristic
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Proof of invariance
Given graphs G1 and G2, find a common refinement H.
Subdivide edges
Add vertices in facesSubdivide faces.
V E F
1 1 0 0
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17
The torus Euler characteristic
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Proof of invariance
Given graphs G1 and G2, find a common refinement H.
Subdivide edges
Add vertices in facesSubdivide faces.
V E F
0 1 1 0
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The torus Euler characteristic
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Proof of invariance
Given graphs G1 and G2, find a common refinement H.
Subdivide edges
Add vertices in facesSubdivide faces.
G1 and H give same G1 and G2 give same
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17
The torus Average degree
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Dont waittriangulate!
We may assume all faces are triangles:
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 11 / 17
The torus Average degree
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Dont waittriangulate!
We may assume all faces are triangles:
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 11 / 17
The torus Average degree
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Count two ways twice
When all faces are triangles:
3F= 2E=
v
degree(v)
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The torus Average degree
C i
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Count two ways twice
When all faces are triangles:
3F= 2E=
v
degree(v)
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 12 / 17
The torus Average degree
A d
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Average degree
V
E+ F = 0 and 3F = 2E=v
degree(v) give
6V = 6E 6F= 6E 4E= 2E
=
v
degree(v)
= 1V
v
degree(v) = 6
= Every triangulation has a vertex of degree at most six
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17
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The torus Average degree
A g d g
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Average degree
V
E+ F = 0 and 3F = 2E=v
degree(v) give
6V = 6E 6F= 6E 4E= 2E
=
v
degree(v)
= 1V
v
degree(v) = 6
= Every triangulation has a vertex of degree at most six
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17
The torus Average degree
Average degree
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Average degree
V
E+ F = 0 and 3F = 2E=v
degree(v) give
6V = 6E 6F= 6E 4E= 2E
=
v
degree(v)
= 1V
v
degree(v) = 6
= Every triangulation has a vertex of degree at most six
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17
The torus Average degree
Average degree
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Average degree
V
E+ F = 0 and 3F = 2E=v
degree(v) give
6V = 6E 6F= 6E 4E= 2E
=
v
degree(v)
= 1V
vdegree(v) = 6
= Every triangulation has a vertex of degree at most six
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17
The torus Average degree
Average degree
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Average degree
V
E+ F = 0 and 3F = 2E=v
degree(v) give
6V = 6E 6F= 6E 4E= 2E
=
v
degree(v)
=1
V
vdegree(v) = 6
= Every triangulation has a vertex of degree at most six
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17
The torus Necessity and sufficiency
Seven suffice
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Seven suffice
Take a vertex-minimal counterexample. . .
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17
The torus Necessity and sufficiency
Seven suffice
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Seven suffice
Take a vertex-minimal counterexample. . .
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17
The torus Necessity and sufficiency
Seven suffice
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Seven suffice
Take a vertex-minimal counterexample. . .
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17
The torus Necessity and sufficiency
Seven suffice
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Seven suffice
Take a vertex-minimal counterexample. . .
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17
The torus Necessity and sufficiency
Seven suffice
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Seven suffice
Take a vertex-minimal counterexample. . .
. . . why, its not a counterexample at all!
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17
The torus Necessity and sufficiency
Seven are necessary
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Seven are necessary
The complete graph K7 embedded on the torus.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 15 / 17
Other surfaces Revisiting the plane
The Four and Five Colour Theorems
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The Four and Five Colour Theorems
Five colours:
A triangulation of the plane has a vertex v of degree at most five.
Kempe chains reduce the number of colours needed for vs
neighbours to four.
Four:
Find an unavoidableset of configurations, and show that none can
occur in a minimal counterexample.
The proof has been simplified by Robinson, Sanders, Seymour
and Thomas (1996), but still requires a computer.
Robinson et. al. use 633 configurations in place of Appel and
Hakens 1476.
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 16 / 17
Other surfaces The Heawood bound
The Heawood bound
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The Heawood bound
Theorem (Heawood, 1890, via average degree arguments))
Maps on a surface of Euler characteristic 1 require at most
7 +
49 22
colours.
The Klein bottle has = 0 but requires only six colours
(Franklin, 1934)Bound is otherwise tight (Ringel and Youngs, 1968)
Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 17 / 17
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