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Fritsch R., Fritsch G. — The Four-Color Theorem: History, Topological Foundations, and Idea of Proof
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Название: The Four-Color Theorem: History, Topological Foundations, and Idea of ProofАвторы: Fritsch R., Fritsch G. Аннотация: This elegant little book discusses a famous problem that helped to define the field now known as graph theory: what is the minimum number of colors required to print a map such that no two adjoining countries have the same color, no matter how convoluted their boundaries are. Many famous mathematicians have worked on the problem, but the proof eluded formulation until the 1970s, when it was finally cracked with a brute-force approach using a computer. The Four-Color Theorem begins by discussing the history of the problem up to the new approach given in the 1990s (by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas). The book then goes into the mathematics, with a detailed discussion of how to convert the originally topological problem into a combinatorial one that is both elementary enough that anyone with a basic knowledge of geometry can follow it and also rigorous enough that a mathematician can read it with satisfaction. The authors discuss the mathematics and point to the philosophical debate that ensued when the proof was announced: just what is a mathematical proof, if it takes a computer to provide one - and is such a thing a proof at all?
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1998Количество страниц: 260Добавлена в каталог: 20.11.2009Операции: Положить на полку |
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Interior point of a set 49 Interior point of an arc 52 Interior vertex of a chain 145 Internally 6-connected 177 Irreducible configuration 168 Irreducible, D 205 Isolated vertex 61 99 140 Isomorphic combinatorial graphs 142 Italy 59 87 Joining arc 51 Jordan closed—curve 47 Jordan curve theorem 51 Jordan curve, associated with a closed chain 153 Jordan, Camille (1838—1922) 47 Kelvin, Lord see “Thomson” Kempe 15—16 17 20 22 59 118 128 131 184 185 Kempe chain 171 Kempe chain game 171 195 200 203 Kempe interchange 172 195 203 Kempe net 172 194 Kempe sector of a boundary coloring 195 Kempe sector of a colored graph 194 Klein, Felix (1849—1925) 21—22 Koch, Helge von (1870—1924) 48 Koch, John 24 35 39 186 215 216 Kummer, Ernst (1810—1893) 19 Kuratowski graphs 143 Kuratowski theorem 144 Kuratowski, Kazimierz (1896—1980) 105 144 L-discharging 229 Large discharging 229 Leg of a configuration 157 Length of a chain 145 Lexicographic ordering 187 Liechtenstein 59 Lindemann, Ferdinand von (1852—1939) 41 Lindley, John (1799—1865) 3 Line graph 111 Line segment 51 Link of a chain 145 Lloyd, E.K. 136 Loop 99 Lorber, Patricia 5 Lyndon, Roger 38 MacKenzie, Donald (* 1950) 34 Maddison, Isabel 22 Major vertex 224 MAP 60 Map, connected 64 Map, cubic 128 Map, dual 121 Map, empty 68 Map, polygonal arc 65 Map, regular 94 Maximal plane graph 105 Maxwell, James Clerk (1831—1879) 118 May, Kenneth Ownsworth 11 Mayer 24 32 32 185 223 228 230 Memory capacity 222 Menasco, William W. 21 Miehe, Friedrich 194 Minimal counterexample 86 Minimal criminal 86 Minimal triangulation 152 Minor vertex 224 MODULO 187 Moebius, August Ferdinand (1790—1868) 21—22 Moise, Edwin E. (* 1918) ix 48 49 51 53 65 Moore, Eliakim Hastings (1863—1932) 25 Morgan see “de Morgan” Multinational comer 46 Multiple edges 99 n-coloring 86 n-coloring of edges 134 n-coloring of vertices (combinatorial) 142 n-coloring, admissible 86 n-gon 52 National border of a map 75 Neighbor of a country 80 Neighbor of a vertex 145 Neighbor, secondary, of a vertex 177 Neighborhood 49 Neighborhood, elementary 67 Neighboring country 80 Neighboring vertices 145 Net see “Kempe net” Neumann, Ernst (1832—1925) 19 Neutral point 61 Neutrality set 62 Newton, Isaac (1643—1727) 20 Normal graph 151 Nowhere dense 48 Obstruction 219 Obstruction, essential 220 Olmsted, John M.H. (1911—1997) 48 Open set 49 Ordering of blocks 207 Ore, Oystein (1899—1968) viii 30 114 185 Osgood, Thomas W. (* 1944) 34 186 Osgood, William Fogg (1864—1943) 48 Outer edge of a configuration 156 Outer vertex of a graph 151 Outset, good from the 192 Parametrization 47 Pared image 28 161 167 Partition 51 Path component 51 Path connected 51 64 Peano curve 48 Peano, Guiseppe (1858—1939) 48 Peirce, Benjamin (1809—1880) 19 Peirce, Charles Sanders 15 18 19 Planar graph 143 Plane graph 141 Poincare, Henri (1854—1912) 26 Point of contact 196 Point, boundary, of a set 49 Point, boundary, of an arc 51 Point, emanating, of a closed curve 52 Point, initial, of an arc 51 Point, interior, of a set 49 Point, interior, of an arc 52 Point, neutral, of a map 61 Point, terminal, of an arc 52 Polygon 52 Polygon, star-shaped 109 Polygonal arc map 65 Polygonal arc without self-intersections 52 Polyhedral formula, Euler 117 Projection, stereographic 45 Proper embedding 194 Qualitative approach 25 Quantitative approach 25 R-discharging 229 Recoloring 203 rectangle 52 Rectilinear triangle 52 Reducer 181 208 Reducibility test 220 Reducible configuration 168 Reducible, A 186 208 210 Reducible, B 186 208 213 Reducible, C 186 208 214 Reducible, D 186—205 215 Regular discharging 229 Regular map 94 Regular solid 117 Regular, dual 120 Reichel, Hans-Christian (* 1945) 109 Reynolds Jr., C.N. 27 185 Right half-open unit interval 47 Ring 153 Ring size 156 Ringel, Gerhard (* 1919) 24 Rinow, Willi (1907—1979) ix 57 Robertson, Neil 41 177 217 Rollet, A.P. 48 Rule of thumb, Heesch's 221 S-discharging 229 Saaty, Thomas L. 176 184 Saeki, Shinichi 36 San Marino 59 87 Sanders, Daniel P. 41 177 217 Saturated graph 105 Schmidt, Erhard (1876—1959) 51 Schmidt, Ulrich 36 Schneider, Ivo (* 1937) 118 Schoenflies, Arthur (1853—1928) theorem 46 53 Secondary neighbor 177 Sector see “Kempe sector” Separated pairs 57 Separating circuit 78 Set of vertices 61 Seymour, Paul D. 41 177 217 Shimamoto, Yoshio (* 1926) 28 Simple chain 145 Simple closed chain 152 Simple closed curve 47 Simple contraction 146 Simple curve 47 Simple subdivision 144 Simply connected 64 Size of a boundary coloring 187 Size of a ring 153 Size, ring 156 Small discharging 229 Smallest criminal 86 Snowflake curve 48 Sommerfeld, Arnold (1868—1951) 29 Spain 87 Spanned subgraph 156 Sparse set of edges 217 Square 52 Stage 202 205 Stanik, Rotraut 186 Star 156 Star point 109 Star shaped 109 Stemple, J.G. 185 Stereographic projection 45 Story 15 18 18 128 Straight arc 51 Stromquist, Walter R. 34 185 Subdivision 144 Subdivision, simple 144 Subgraph 141 Subgraph, spanned 156 Subtraction modulo r 187 Supply problem 143 Swart, Edward Reinier 34 217 Switzerland 59 Sylvester 14 15 18 19 Szele, Tibor 105 T-discharging 229 Tait 20 20 22 118 134—137 217 Terminal point 52 Thistlethwaite, Morwen B. 21 Thomas, Robin 41 177 217 Thomson, William (1824—1907) 20 Tlitte, William Thomas (* 1917) 25 33 34 40 137 156 199 Transversal discharging 229 TREE 69 199 triangle 52 69 153 Triangle, curvilinear 101 Triangle, rectilinear 52 Triangulated graph 105 Ubesgue, Henri (1875—1941) 27 Unavoidable set 168 Unbounded face 105 Unbounded set 44 Underlying combinatorial graph 142 Unit circle 50 Unit disk 64 Unit interval 47 Unit interval, closed 47 Unit interval, right half-open 47 United States of America 45 Valery, Paul (1871—1945) 32 Veblen, Oswald (1880—1960) 25 Vertex 61 Vertex coloring 139 Vertex coloring (combinatorial) 142 Vertex coloring, admissible 139 Vertex coloring, admissible (combinatorial) 142 Vertex n-coloring 139 Vertex n-coloring (combinatorial) 142 Vertex of degree d 88 Vertex, final, of a map 69 Vertex, inner, of a graph 151 Vertex, interior, of a chain 145 Vertex, isolated 61 99 140 Vertex, major 224 Vertex, minor 224 Vertex, neighboring 145 Vertex, outer, of a graph 151 Wagner and Fary theorem 111 Wagner, Klaus (* 1910) 61 105 109 114 144 145 150 Weierstrass, Karl (1815—1897) 19 Weise, Karl Heinrich (1909—1990) 36 Weiske Theorem 125 Weiske, Benjamin Gotthold (1783—1836) 21—22 185 Wernicke, Paul (1866—19??) 167 169 224 Weyl, Hermann (1885—1955) 29 Wheel 156 Whewell 10—11 12 Whitney, Hassler (1907—1989) 34 156 199 Wilson, R.J. 136 Winding number 51 Winn, C.E. 27 185 186 Youngs, J.W.T. 24
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