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The Notorious Four-Color Problem

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The Notorious Four-Color Problem 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 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs The Map-Coloring Problem Question: How many colors are required to color a map of the United States so that no two adjacent regions are given the same color? Answer: Four colors are enough. Three are not enough. The Four-Color Problem: Is there some map that requires five colors? In order to give a negative answer, you have to show that every map | no matter how cleverly constructed | can be colored with 4 or fewer colors. 2 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs 3 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs 4 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs 5 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs 6 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs The History of the Four-Color Theorem 1852: Student Francis Guthrie notices that four colors suffice to color a map of the counties of England. Guthrie poses the Four-Color Problem to his brother Frederick, a student of Augustus De Morgan (a big shot of 19th-century British mathematics). De Morgan likes the problem and mentions it to others. 7 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs The History of the Four-Color Theorem I 1879: Alfred Kempe proves the Four-Color Theorem (4CT): Four colors suffice to color any map. I 1880: Peter Tait finds another proof. That was that. I 1890: Percy John Heawood shows that Kempe's proof was wrong. I 1891: Julius Petersen shows that Tait's proof was wrong. I 20th century: Many failed attempts to (dis)prove the 4CT. Some lead to interesting discoveries; many don't. 8 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs The History of the Four-Color Theorem Mathematician H.S.M. Coxeter: Almost every mathematician must have experienced one glorious night when he thought he had discovered a proof, only to find in the morning that he had fallen into a similar trap. Mathematician Underwood Dudley: The four-color conjecture was easy to state and easy to understand, no large amount of technical mathematics is needed to attack it, and errors in proposed proofs are hard to see, even for professionals; what an ideal combination to attract cranks! 9 / 60 The Four-Color Theorem Graphs Coloring Maps The Solution of the Four-Color Problem History More About Coloring Graphs The History of the Four-Color Theorem I 1976: Kenneth Appel and Wolfgang Haken prove the 4CT. Their proof relies on checking a large number of cases by computer, sparking ongoing debate over what a proof really is. I 1997: N. Robertson, D.P. Sanders, P.D. Seymour, and R. Thomas improve Appel and Haken's methods to reduce the number of cases (but still rely on computer assistance). I 2005: Georges Gonthier publishes a \formal proof" (automating not just the case-checking, but the proof process itself). 10 / 60 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 A graph consists of a collection of vertices connected by edges. vertices edges 11 / 60 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 The vertices and edges of a graph do not have to be points and curves. I Facebook: vertices = people, edges = friendship I WWW: vertices = web pages, edges = links I Chess: vertices = positions, edges = possible moves 12 / 60 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 vs. Drawings A graph remains the same no matter how you draw it. (a) 2 (b) 2 3 1 5 5 1 4 4 3 Note: crossings (like this) are not vertices. These drawings represent the same graph (e.g., vertex 5 has neighbors 2,4 in both cases). All that matters is which pairs of vertices are connected. 13 / 60 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. (a) Planar 2 (b) Planar (same graph as (a)) 2 3 1 5 5 1 4 4 2 3 1 The key word in the 3 5 definition is "can". (c) Not planar 4 14 / 60 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. 15 / 60 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 16 / 60 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.) The Four-Color Problem: Does every planar graph have chromatic number 4 or less? 17 / 60 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 = 4 χ = 3 χ = 3 χ = 2 χ = 3 χ = 2 χ = 5 χ = 2χ = 4 18 / 60 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! 19 / 60 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? I 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"). I When is a proof correct? | The standard of proof is very high in mathematics (not just \beyond a reasonable doubt", but beyond any doubt) I Who gets to decide whether a proof is correct? I Are some proofs better than others? 20 / 60 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 21 / 60 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: 22 / 60 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 sea field #6 sea field #5 field #4 sea One by one, the dikes break under the pressure.
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