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

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    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

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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?

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    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?

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    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!).

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    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

    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!

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    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??

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    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.

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    The torus From maps to graphs

    http://find/http://goback/
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    p g p

    From maps to graphs

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    The torus From maps to graphs

    http://find/http://goback/
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    p g p

    From maps to graphs

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    The torus From maps to graphs

    http://goforward/http://find/
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    From maps to graphs

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    The torus From maps to graphs

    http://find/http://goback/
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    From maps to graphs

    The dual of the map

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    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

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    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.

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    The torus Euler characteristic

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

    (torus) = 1 2 + 1 = 0 (sphere) = 4 6 + 4 = 2

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    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.

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    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

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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

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    The torus Average degree

    http://goforward/http://find/http://goback/
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    Dont waittriangulate!

    We may assume all faces are triangles:

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    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)

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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)

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    The torus Average degree

    A d

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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    Seven suffice

    Take a vertex-minimal counterexample. . .

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    The torus Necessity and sufficiency

    Seven suffice

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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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

    http://find/
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    Seven are necessary

    The complete graph K7 embedded on the torus.

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    Other surfaces Revisiting the plane

    The Four and Five Colour Theorems

    http://find/
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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.

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    Other surfaces The Heawood bound

    The Heawood bound

    http://find/
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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

    http://find/