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

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 Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 2 / 48

The Four Colour Theorem The proof Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 3 / 48

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

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

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

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

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

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

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

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

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

The Four Colour Theorem The proof first proof (attempt) by Kempe in 1879 Sebastian Wheeler Four Colour Theorem 11 / 48

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

The Four Colour Theorem Structure of the proof The proof Unavoidability Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 12 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Definition (polygonal outline, polygonal map, face) A polygonal outline is the pairwise disjoint union of a finite number of open 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

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: A general polygonal map Sebastian Wheeler Four Colour Theorem 14 / 48

The Four Colour Theorem Structure of the proof 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

The Four Colour Theorem Structure of the proof 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

The Four Colour Theorem Structure of the proof The proof Unavoidability Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 16 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 17 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Reducing to cubic map Sebastian Wheeler Four Colour Theorem 18 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Reducing to cubic map 3 V = 2 E Sebastian Wheeler Four Colour Theorem 18 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Reducing to cubic map 3 V = 2 E V − E + F = 2 Sebastian Wheeler Four Colour Theorem 18 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Reducing to cubic map 3 V = 2 E V − E + F = 2 2 E = 6 F − 12 Sebastian Wheeler Four Colour Theorem 18 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Reducing to cubic map 3 V = 2 E V − E + F = 2 2 E F = 6 − 12 2 E = 6 F − 12 F Sebastian Wheeler Four Colour Theorem 18 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 19 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Possible shape and neighbourhood of the smaller face Sebastian Wheeler Four Colour Theorem 20 / 48

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Kempe chain Sebastian Wheeler Four Colour Theorem 35 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Kempe chain Sebastian Wheeler Four Colour Theorem 36 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Kempe chains splitting the colours Sebastian Wheeler Four Colour Theorem 37 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Counterexample to Kempe’s proof Sebastian Wheeler Four Colour Theorem 38 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Counterexample to Kempe’s proof Sebastian Wheeler Four Colour Theorem 39 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Figure: Counterexample to Kempe’s proof Sebastian Wheeler Four Colour Theorem 40 / 48

The Four Colour Theorem Structure of the proof The proof Unavoidability Table of Contents The Four Colour Theorem 1 The proof 2 Structure of the proof Simplify the map Consider a minimal counterexample Correct Kempe’s mistake Unavoidability Sebastian Wheeler Four Colour Theorem 41 / 48