Four Colour Theorem Sebastian Wheeler Technische Universit at M - - PowerPoint PPT Presentation

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Four Colour Theorem Sebastian Wheeler Technische Universit at M - - PowerPoint PPT Presentation

Aug 09, 2023 231 likes •870 views

The Four Colour Theorem The proof Four Colour Theorem Sebastian Wheeler Technische Universit at M unchen 19.06.2018 Sebastian Wheeler Four Colour Theorem 1 / 48 The Four Colour Theorem The proof Table of Contents The Four Colour

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The Four Colour Theorem The proof

Four Colour Theorem

Sebastian Wheeler

Technische Universit¨ at M¨ unchen

19.06.2018

Sebastian Wheeler Four Colour Theorem 1 / 48

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The Four Colour Theorem The proof

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 2 / 48

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The Four Colour Theorem The proof

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 3 / 48

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The Four Colour Theorem The proof Sebastian Wheeler Four Colour Theorem 4 / 48

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The Four Colour Theorem The proof Sebastian Wheeler Four Colour Theorem 5 / 48

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The Four Colour Theorem The proof Sebastian Wheeler Four Colour Theorem 6 / 48

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The Four Colour Theorem The proof Sebastian Wheeler Four Colour Theorem 7 / 48

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The Four Colour Theorem The proof Sebastian Wheeler Four Colour Theorem 8 / 48

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The Four Colour Theorem The proof

Theorem (Four Colour Theorem) The regions of any simple planar map can be coloured with only four colours, in such a way that any two adjacent regions have different colours.

Sebastian Wheeler Four Colour Theorem 9 / 48

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The Four Colour Theorem The proof

Theorem (Four Colour Theorem) The regions of any simple planar map can be coloured with only four colours, in such a way that any two adjacent regions have different colours. Definition (Planar map, regions, simple map) A planar map is a set of pairwise disjoint subsets of the plane, called

  • regions. A simple map is one whose regions are connected open sets.

Sebastian Wheeler Four Colour Theorem 9 / 48

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The Four Colour Theorem The proof

Definition (Adjacent regions) Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map.

Sebastian Wheeler Four Colour Theorem 10 / 48

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The Four Colour Theorem The proof

Definition (Adjacent regions) Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. Definition (Corner) A point is a corner of a map iff it belongs to the closures of at least three regions.

Sebastian Wheeler Four Colour Theorem 10 / 48

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The Four Colour Theorem The proof

first proof (attempt) by Kempe in 1879

Sebastian Wheeler Four Colour Theorem 11 / 48

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The Four Colour Theorem The proof

first proof (attempt) by Kempe in 1879 correct proof by Appel and Haken in 1976

Sebastian Wheeler Four Colour Theorem 11 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 12 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Definition (polygonal outline, polygonal map, face) A polygonal outline is the pairwise disjoint union of a finite number of

  • pen line segments, called edges, with a set of nodes that contains the

endpoints of the edges. The regions of a finite polygonal planar map, called faces, are the connected components of the complement of a polygonal outline.

Sebastian Wheeler Four Colour Theorem 13 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: A general polygonal map

Sebastian Wheeler Four Colour Theorem 14 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: A general polygonal map

Definition (polyhedral map) A polyhedral map is a finite bridgeless connected polygonal map.

Sebastian Wheeler Four Colour Theorem 14 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Theorem (Euler formula) The Euler polyhedron formula (Euler 1752) V − E + F = 2 where V,E,F are the number of vertices, edges and faces, is valid for any schema which represents the sphere.

Sebastian Wheeler Four Colour Theorem 15 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 16 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 17 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Reducing to cubic map

Sebastian Wheeler Four Colour Theorem 18 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Reducing to cubic map

3V = 2E

Sebastian Wheeler Four Colour Theorem 18 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Reducing to cubic map

3V = 2E V − E + F = 2

Sebastian Wheeler Four Colour Theorem 18 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Reducing to cubic map

3V = 2E V − E + F = 2 2E = 6F − 12

Sebastian Wheeler Four Colour Theorem 18 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Reducing to cubic map

3V = 2E V − E + F = 2 2E = 6F − 12 2E F = 6 − 12 F

Sebastian Wheeler Four Colour Theorem 18 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 19 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Possible shape and neighbourhood of the smaller face

Sebastian Wheeler Four Colour Theorem 20 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 21 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 22 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 23 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 24 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 25 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 26 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 27 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 28 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 29 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 30 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 31 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 32 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 33 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability Sebastian Wheeler Four Colour Theorem 34 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Kempe chain

Sebastian Wheeler Four Colour Theorem 35 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Kempe chain

Sebastian Wheeler Four Colour Theorem 36 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Kempe chains splitting the colours

Sebastian Wheeler Four Colour Theorem 37 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Counterexample to Kempe’s proof

Sebastian Wheeler Four Colour Theorem 38 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Counterexample to Kempe’s proof

Sebastian Wheeler Four Colour Theorem 39 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Counterexample to Kempe’s proof

Sebastian Wheeler Four Colour Theorem 40 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 41 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

The task is to find a certain set of configurations that are:

Sebastian Wheeler Four Colour Theorem 42 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

The task is to find a certain set of configurations that are: reducible, meaning that it can be shown that they are 4-colourable and

Sebastian Wheeler Four Colour Theorem 42 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

The task is to find a certain set of configurations that are: reducible, meaning that it can be shown that they are 4-colourable and unavoidable, meaning that one of these configurations has to appear in every counterexample considered.

Sebastian Wheeler Four Colour Theorem 42 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Table of Contents

1

The Four Colour Theorem

2

The proof Structure of the proof

Simplify the map Consider a minimal counterexample Correct Kempe’s mistake

Unavoidability

Sebastian Wheeler Four Colour Theorem 43 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharging

i(f ) = 6 − #sides(f )

Sebastian Wheeler Four Colour Theorem 44 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharging

i(f ) = 6 − #sides(f )

  • f ∈F

i(f ) = 6|F| − 2|E| = 6|F| − 6|F| + 12 = 12

Sebastian Wheeler Four Colour Theorem 44 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge rules

Discharge Rule Do nothing.

Sebastian Wheeler Four Colour Theorem 45 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge rules

Discharge Rule Do nothing. Discharge Rule Transfer 1

5 from every pentagon to every adjacent face with more than six

sides.

Sebastian Wheeler Four Colour Theorem 45 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge rules

Discharge Rule Do nothing. Discharge Rule Transfer 1

5 from every pentagon to every adjacent face with more than six

sides. r(fk) = (6 − k) + m 5 ≤ (6 − k) + k 5 = 30 − 5k + k 5 ≤ −2 5 < 0

Sebastian Wheeler Four Colour Theorem 45 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: First unavoidable set

Sebastian Wheeler Four Colour Theorem 46 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge Rule Transfer 1

4 from every pentagon to up to 4 adjacent faces with more than

six sides.

Sebastian Wheeler Four Colour Theorem 47 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge Rule Transfer 1

4 from every pentagon to up to 4 adjacent faces with more than

six sides. r(f7) = −1 + m 4

Sebastian Wheeler Four Colour Theorem 47 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Discharge Rule Transfer 1

4 from every pentagon to up to 4 adjacent faces with more than

six sides. r(f7) = −1 + m 4 r(fk) = (6 − k) + m 4 ≤ 24 − 3k 4 ≤ 0

Sebastian Wheeler Four Colour Theorem 47 / 48

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The Four Colour Theorem The proof Structure of the proof Unavoidability

Figure: Second, larger unavoidable set

Sebastian Wheeler Four Colour Theorem 48 / 48