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

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

Outline Introduction 1 Map colouring The torus 2 From maps to graphs Euler characteristic Average degree Necessity and sufficiency Other surfaces 3 Revisiting the plane The Heawood bound Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 2 / 17

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

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

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

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

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

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

Introduction Map colouring On the donut they do nut! Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17

Introduction Map colouring On the donut they do nut! Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 5 / 17

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 The Seven Colour Theorem Theorem Seven colours are necessary and sufficient to properly colour maps on a torus. Steps: Simplify! 1 Use the Euler characteristic to find the average degree . 2 Look at a minimal counterexample. 3 Prove necessity. 4 Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 6 / 17

The torus From maps to graphs From maps to graphs Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

The torus From maps to graphs From maps to graphs Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

The torus From maps to graphs From maps to graphs Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 7 / 17

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

The torus Euler characteristic Euler characteristic S a surface G a graph drawn on S so that no edges or vertices cross or 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

The torus Euler characteristic Euler characteristic S a surface �������� �������� G a graph drawn on S so that �������� �������� �������� �������� �������� �������� no edges or vertices cross �������� �������� or 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

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

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , 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

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , find a common refinement H . Subdivide edges Add vertices in faces Subdivide faces. ∆ V ∆ E ∆ F ∆ χ 1 1 0 0 Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , 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

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , find a common refinement H . Subdivide edges Add vertices in faces Subdivide faces. ∆ V ∆ E ∆ F ∆ χ 1 1 0 0 Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , find a common refinement H . Subdivide edges Add vertices in faces Subdivide faces. ∆ V ∆ E ∆ F ∆ χ 0 1 1 0 Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

The torus Euler characteristic Proof of invariance Given graphs G 1 and G 2 , find a common refinement H . Subdivide edges Add vertices in faces Subdivide faces. ⇒ G 1 and H give same χ ⇒ G 1 and G 2 give same χ Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 10 / 17

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

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

The torus Average degree Count two ways twice When all faces are triangles: � 3 F = 2 E = degree ( v ) v Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 12 / 17

The torus Average degree Count two ways twice When all faces are triangles: � 3 F = 2 E = degree ( v ) v Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 12 / 17

The torus Average degree Count two ways twice When all faces are triangles: � 3 F = 2 E = degree ( v ) v Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 12 / 17

The torus Average degree Average degree � V − E + F = 0 and 3 F = 2 E = degree ( v ) give v 6 V = 6 E − 6 F = 6 E − 4 E = 2 E � = degree ( v ) v 1 � = degree ( v ) = 6 ⇒ V v = 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 � V − E + F = 0 and 3 F = 2 E = degree ( v ) give v 6 V = 6 E − 6 F = 6 E − 4 E = 2 E � = degree ( v ) v 1 � = degree ( v ) = 6 ⇒ V v = 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 � V − E + F = 0 and 3 F = 2 E = degree ( v ) give v 6 V = 6 E − 6 F = 6 E − 4 E = 2 E � = degree ( v ) v 1 � = degree ( v ) = 6 ⇒ V v = Every triangulation has a vertex of degree at most six ⇒ Christopher Tuffley (Massey University) The Seven Colour Theorem November 2008 13 / 17