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

    http://find/
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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

    Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 2 / 17

    http://find/http://goback/
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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

    http://find/
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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

    http://find/
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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

    http://find/http://goback/
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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

    http://find/http://goback/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/http://goback/
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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

    http://find/http://goback/
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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

    http://goforward/http://find/
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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

    http://find/http://goback/
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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

    http://find/http://goback/
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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

    http://find/http://goback/
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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

    http://find/http://goback/
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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

    http://find/
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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

    http://find/http://goback/
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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

    http://find/
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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

    http://find/http://goback/
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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

    http://goforward/http://find/http://goback/
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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

    http://goforward/http://find/http://goback/
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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

    http://find/
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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

    http://find/
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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

    http://find/