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The Seven Colour Theorem Christopher Tuffley Institute of - - PowerPoint PPT Presentation

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The Seven Colour Theorem Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North 3rd Annual NZMASP Conference, November 2008 Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 1 /

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SLIDE 1

The Seven Colour Theorem

Christopher Tuffley

Institute of Fundamental Sciences Massey 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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SLIDE 2

Outline

1

Introduction Map colouring

2

The torus From maps to graphs Euler characteristic Average degree Necessity and sufficiency

3

Other surfaces Revisiting the plane The Heawood bound

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

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SLIDE 3

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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SLIDE 4

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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SLIDE 5

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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SLIDE 6

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

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SLIDE 7

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

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SLIDE 8

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

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SLIDE 9

Introduction Map colouring

On the donut they do nut!

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17

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SLIDE 10

Introduction Map colouring

On the donut they do nut!

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17

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SLIDE 11

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

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SLIDE 12

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 characteristic to 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

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SLIDE 13

The torus From maps to graphs

From maps to graphs

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

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SLIDE 14

The torus From maps to graphs

From maps to graphs

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

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SLIDE 15

The torus From maps to graphs

From maps to graphs

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

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SLIDE 16

The torus From maps to graphs

From maps to graphs

The dual of the map

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

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SLIDE 17

The torus Euler characteristic

Euler characteristic

S a surface G a graph drawn on S so that

no edges or vertices cross

  • r overlap

all regions (faces) are discs there are V vertices, E edges, F faces.

Definition The Euler characteristic of S is χ(S) = V − E + F. Theorem χ(S) depends only on S and not on G.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 8 / 17

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SLIDE 18

The torus Euler characteristic

Euler characteristic

S a surface G a graph drawn on S so that

no edges or vertices cross

  • r overlap

all regions (faces) are discs there are V vertices, E edges, F faces.

  • Definition

The Euler characteristic of S is χ(S) = V − E + F. Theorem χ(S) depends only on S and not on G.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 8 / 17

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SLIDE 19

The torus Euler characteristic

Examples

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

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 9 / 17

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The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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SLIDE 21

The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces. ∆V ∆E ∆F ∆χ 1 1

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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SLIDE 22

The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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SLIDE 23

The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces. ∆V ∆E ∆F ∆χ 1 1

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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SLIDE 24

The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces. ∆V ∆E ∆F ∆χ 1 1

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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SLIDE 25

The torus Euler characteristic

Proof of invariance

Given graphs G1 and G2, find a common refinement H. Subdivide edges Add vertices in faces Subdivide faces. ⇒ G1 and H give same χ ⇒ G1 and G2 give same χ

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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The torus Average degree

Don’t wait—triangulate!

We may assume all faces are triangles:

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 11 / 17

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SLIDE 27

The torus Average degree

Don’t wait—triangulate!

We may assume all faces are triangles:

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 11 / 17

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The torus Average degree

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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SLIDE 29

The torus Average degree

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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SLIDE 30

The torus Average degree

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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SLIDE 31

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 32

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 33

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 34

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 35

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 36

The torus Average degree

Average degree

V − E + F = 0 and 3F = 2E =

  • v

degree(v) give 6V = 6E − 6F = 6E − 4E = 2E =

  • v

degree(v) = ⇒ 1 V

  • 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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SLIDE 37

The torus Necessity and sufficiency

Seven suffice

Take a vertex-minimal counterexample. . .

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17

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SLIDE 38

The torus Necessity and sufficiency

Seven suffice

Take a vertex-minimal counterexample. . .

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17

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SLIDE 39

The torus Necessity and sufficiency

Seven suffice

Take a vertex-minimal counterexample. . .

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17

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SLIDE 40

The torus Necessity and sufficiency

Seven suffice

Take a vertex-minimal counterexample. . .

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17

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SLIDE 41

The torus Necessity and sufficiency

Seven suffice

Take a vertex-minimal counterexample. . . . . . why, it’s not a counterexample at all!

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 14 / 17

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The torus Necessity and sufficiency

Seven are necessary

The complete graph K7 embedded on the torus.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 15 / 17

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Other surfaces Revisiting the plane

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 v’s neighbours to four. Four: Find an unavoidable set of configurations, and show that none can

  • ccur 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 Haken’s 1476.

Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 16 / 17

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Other surfaces The Heawood bound

The Heawood bound

Theorem (Heawood, 1890, via average degree arguments)) Maps on a surface of Euler characteristic χ ≤ 1 require at most 7 + √49 − 2χ 2

  • colours.

The Klein bottle has χ = 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