The Notorious Four-Color Problem Prof. Jeremy L. Martin Department - PowerPoint PPT Presentation

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Planar Graphs A graph is planar if its vertices and edges can be drawn as points and line segments with no crossings. 33 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Planar Graphs A graph is planar if its vertices and edges can be drawn as points and line segments with no crossings. 2 (b) Planar (same graph as (a)) (a) Planar 2 3 5 1 5 1 4 4 2 3 1 3 5 (c) Not planar 4 34 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Planar Graphs A graph is planar if its vertices and edges can be drawn as points and line segments with no crossings. 2 (b) Planar (same graph as (a)) (a) Planar 2 3 5 1 5 1 4 4 2 3 1 The key word in the 3 definition is "can". 5 (c) Not planar 4 35 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graphs from Maps Every map can be modeled as a planar graph. Vertices represent regions; edges represent common borders. 36 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring Example: You are a kindergarten teacher. You want to assign each child a table to sit at. However, there are certain pairs of kids who shouldn’t sit together. How many tables are you going to need? 40 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring Example: You are a kindergarten teacher. You want to assign each child a table to sit at. However, there are certain pairs of kids who shouldn’t sit together. How many tables are you going to need? This is a graph theory problem! 41 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring Example: You are a kindergarten teacher. You want to assign each child a table to sit at. However, there are certain pairs of kids who shouldn’t sit together. How many tables are you going to need? This is a graph theory problem! vertices = children edges = pairs of kids to keep separate colors = tables 42 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring and the Chromatic Number A proper coloring of G is an assignment of colors to the vertices of G such that every two vertices connected by an edge must receive different colors. The chromatic number χ ( G ) of G is the minimum number of colors needed for a proper coloring. “ G is k -colorable” means “the chromatic number of G is k or less”. 43 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring and the Chromatic Number χ χ χ χ = 4 = 3 = 3 = 2 χ χ = 2 = 3 χ χ χ = 5 = 2 = 4 44 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring and the Chromatic Number A proper coloring of G is an assignment of colors to the vertices of G such that every two vertices connected by an edge must receive different colors. The chromatic number χ ( G ) is the minimum number of colors needed for a proper coloring. Important Note: The chromatic number is not necessarily the same as the maximum number of mutually connected vertices. (For example, a graph can have chromatic number 3 even if it has no triangles.) 45 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring Consider a graph whose vertices are a 9 × 9 grid of points. Two vertices are joined by an edge if they are in the same row, column, or 3 × 3 subregion. Not all edges shown − each vertex has 20 neighbors. You are given a partial proper coloring and told to extend it to all vertices. Does this sound familiar? 46 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring Consider a graph whose vertices are a 9 × 9 grid of points. Two vertices are joined by an edge if they are in the same row, column, or 3 × 3 subregion. Not all edges shown − each vertex has 20 neighbors. You are given a partial proper coloring and told to extend it to all vertices. Does this sound familiar? It’s a Sudoku puzzle! 47 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Graph Coloring A proper coloring of G is an assignment of colors to the vertices of G such that every two vertices connected by an edge must receive different colors. The chromatic number χ ( G ) of G is the minimum number of colors needed for a proper coloring. The Four-Color Problem: Does every planar graph have chromatic number 4 or less? 48 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Interlude: What Is A Proof? ◮ What is a mathematical proof? — A logical argument that relies on commonly accepted axioms and rules of inference (e.g., “if a = b and b = c , then a = c ”). ◮ When is a proof correct? — The standard of proof is very high in mathematics (not just “beyond a reasonable doubt”, but beyond any doubt) ◮ Who gets to decide whether a proof is correct? ◮ Are some proofs better than others? 49 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Back to Graph Theory: Faces of Planar Graphs Planar graphs have faces as well as edges and vertices. The faces are the areas between the edges. v = 8 v = 5 e = 12 e = 8 f = 6 f = 5 v = 13 v = 6 e = 19 e = 12 f = 8 f = 8 50 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula Theorem (Euler’s Formula) Let G be any planar graph. Let v, e, f denote the numbers of vertices, edges, and faces, respectively. Then, v − e + f = 2 . 51 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof Imagine that the edges are dikes that hold back the raging sea from a network of fields. sea sea sea field #2 field #3 field #6 sea sea field #5 field #4 sea One by one, the dikes break under the pressure. Each time a dike breaks, the raging sea rushes into one of the fields, and there is one fewer field than before. 52 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof sea sea sea v = 12 e = 16 f = 6 sea sea sea 53 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof Finally, the raging sea has overwhelmed all of the enclosed areas — the graph has only one face There are still some dikes left, but they’re now just piers extending into the sea. One by one, the network of piers shrinks. 60 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof ◮ After all the dikes (= edges) are gone, v = 1 , e = 0 , f = 1 , v − e + f = 2 . 73 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof ◮ After all the dikes (= edges) are gone, v = 1 , e = 0 , f = 1 , v − e + f = 2 . ◮ Each time a dike disappeared, either f or v decreased by 1. 74 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof ◮ After all the dikes (= edges) are gone, v = 1 , e = 0 , f = 1 , v − e + f = 2 . ◮ Each time a dike disappeared, either f or v decreased by 1. ◮ Therefore, the value of v − e + f never changed — it must always have been 2! 75 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Euler’s Formula: The “Raging Sea” Proof ◮ After all the dikes (= edges) are gone, v = 1 , e = 0 , f = 1 , v − e + f = 2 . ◮ Each time a dike disappeared, either f or v decreased by 1. ◮ Therefore, the value of v − e + f never changed — it must always have been 2! ◮ This logic works no matter what the original graph was — so we have proved Euler’s formula for all planar graphs. 76 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Proofs and The Book Most mathematicians would call this proof “beautiful” or a “proof from the Book” — it is a simple and elegant proof of an extremely important theorem. The mathematician Paul Erd˝ os spoke of “The Book,” in which God records the best and most beautiful proofs of mathematical theorems. Once in a while, a mortal is allowed a glimpse of the Book. Erd˝ os said: “You don’t have to believe in God, but you should believe in the Book.” 77 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Degrees and Lengths First, a little more terminology. The degree d ( V ) of a vertex v in a graph G is the number of edges having v as an endpoint. The length ℓ ( F ) of a face F in a planar graph is the number of edges around it. (Edges that poke into F count double.) 78 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Degrees and Lengths 4 2 8 6 1 3 blue: 3 face lengths 2 6 3 red: 4 vertex degrees 4 3 2 3 2 4 79 / 141

The Four-Color Theorem Definitions and Examples Graphs Planar Graphs The Solution of the Four-Color Problem Coloring Graphs More About Coloring Graphs Euler’s Formula Degrees, Lengths, and Euler’s Formula Theorem 1 (“Handshaking”): In a graph with e edges, the sum of the degrees of all vertices is 2 e . (Reason: Every edge contributes to the degrees of two vertices.) Theorem 2: In a planar graph with e edges, the sum of the lengths of all faces is 2 e . ◮ In a planar graph, each face has length 3 or greater, so 2 e ≥ 3 f , so f ≤ 2 e / 3. ◮ By Euler’s formula, v − e + f = 2 ≤ v − e + 2 e / 3 = v − e / 3. ◮ Applying a little algebra to 2 ≤ v − e / 3 gives us e ≤ 3 v − 6. 80 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments The Six-Color Theorem Theorem Every planar graph can be colored with 6 or fewer colors. Idea: Find the right order in which to color the vertices so that we will never need more than 6 colors. ◮ The trick is to think about which vertex to color last. ◮ It should be a vertex with at most 5 neighbors. ◮ How do we know G has such a vertex? 81 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments The Six-Color Theorem Suppose G is a planar graph. ◮ If every vertex of G has degree 6 or greater, then the sum of degrees would be at least 6 v , and there would be at least 3 v edges. ◮ But this is impossible since e ≤ 3 v − 6. ◮ Therefore, there must be some vertex whose degree is 5 or less. (In fact, there will be many choices — but we only need one.) This is the vertex we are going to color last. 82 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments Successively Deleting Vertices of Degree 6 Or Less 83 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments Successively Deleting Vertices of Degree 6 Or Less degree 3 84 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments Successively Deleting Vertices of Degree 6 Or Less Et cetera. 93 / 141

The Four-Color Theorem The Six-Color Theorem Graphs Appel and Haken’s Proof The Solution of the Four-Color Problem Reactions to Appel and Haken’s Proof More About Coloring Graphs Recent Developments Color the vertices in reverse order — last deleted = first colored. Always use the first possible color from the palette. 10 12 3 8 7 11 14 Palette 2 5 13 6 4 1 9 94 / 141

The Notorious Four-Color Problem Prof. Jeremy L. Martin Department - PowerPoint PPT Presentation

The Four-Color Theorem Graphs The Solution of the Four-Color Problem More About Coloring Graphs The Notorious Four-Color Problem Prof. Jeremy L. Martin Department of Mathematics University of Kansas KU Mini College June 5, 2013 1 / 141

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The Notorious Four-Color Problem Prof. Jeremy L. Martin Department - PowerPoint PPT Presentation

The Four-Color Theorem Graphs The Solution of the Four-Color Problem More About Coloring Graphs The Notorious Four-Color Problem Prof. Jeremy L. Martin Department of Mathematics University of Kansas KU Mini College June 5, 2013 1 / 141

Color and Color Models Werner Purgathofer Color problem specification light and perception

COLOR (or colour) Color Perception Visible color Gamut RGB color Gamut Pantone color Gamut

1 The Hum an Eye The Hum an Eye The eye: The center of the retina is a densely packed region

Selling IQ Color Solutions February 2018 IQ Color Overview Why Use Color? IQ Color

Color and Icons SWEN-444 Color Topics Color perception Using color in interaction design

together, the entire spectrum of color is absorbed, and were left with black. Primaries: Red,

Lecture 2: Color Tuesday, Sept 4 Color Radiometry: some definitions Color of light arriving

Color Management and Your Workflow monaco by Problem in Matching Colors &gt; THE RESULTS

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Forensic Techniques that Helped Convict Some of the Worlds Most Notorious Serial Killers

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DESIGN REVIEW 1.2 - Notorious EMG Chris Anderson (EE), Jacob Gamboa (EE), Marshall Kabat (ME), Vi

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Color superfluidity of neutral ultracold fermions in the presence of color-orbit and color-flip

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Sector call - Efficient Natural Resources 7 th March 2018 Jane Toogood Sector Chief Executive,

Adaptable colouring and colour critical graphs Bing Zhou Department of Mathematics Trent

On choosability of graphs with limited number of colours M. Demange, RMIT University,

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353,

Counting colour symmetries of regular tilings Dirk Frettl oh University of Bielefeld

The Seven Colour Theorem Christopher Tuffley Institute of Fundamental Sciences Massey

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