# Seven Colours

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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

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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

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 9 / 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.

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

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.

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)

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 12 / 17

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