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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 Proof
Авторы: 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?
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 1998
Количество страниц: 260
Добавлена в каталог: 20.11.2009
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Предметный указатель
-chain 145
-good 203
-compatible boundary coloring 210
-proper embedding 209
4-adic number 190
4-gon, complete 101
6-6-chain 229
A-reducible 186 208 210
Abutting blocks 195
Accessible, arcwise 48
Addition modulo r 187
Adjacency matrix 192
Admissible coloring 44
Admissible n-coloring 86
Admissible vertex coloring 139
Admissible vertex coloring (combinatorial) 142
Aigner, Martin (* 1942) 24 135 137 144 176 222
Allaire, Frank 34 186 216 217
Andorra 87
Andrews, Thomas (1813—1885) 20
Appel 24 30 33—37 37 85 148 183 186 192 215 216 222—223 228—230
Approach, qualitative 25
Approach, quantitative 25
ARC 47
Arc, associated with a chain 153
Arc, joining 51
Arc, straight 51
Arcwise accessible 48
Arnold, Wolfgang (* 1930) 118
Articulation 161 219
Associated arc 153
Associated Jordan curve 153
Austria 59
B-reducible 186 208 213
Bacon, Francis (1560—1626) 13
Balloon 44
Baltzer, Richard (1818—1887) 21—22
Bemhart, Arthur 27 34 183
Bemhart, Frank R. 34
Berber, Felix (1871—1930) 29
Berge, Claude 33
Bigaike, Hans-Guenther (* 1933) 30 186 201
Biggs, Norman L. 136
Birkhoff 25 128 153 176 184—186 210
Birkhoff diamond 156 187—214
Birkhoff number 184—185
Block 195
Block decomposition 195
block selection 203
Block, abutting 195
Block, ordering of 207o
Bocher, Maxime (1867—1918) 25
Bodendiek, Rainer (* 1940) 61 114 144 145 150
Bolus, Harry (1834—1911) 4—5
Boole, George (1815—1864) 10
Border of a country 44 75
Border point of a country 73
Border, national, of a map 75
Border-stone 60
Borderline 46
Borderline of a country 75
Borel, Emile (1871—1956) 59
Boundary coloring 187
Boundary coloring of a configuration 190
Boundary coloring, -good 203
Boundary coloring, -compatible 210
Boundary coloring, equivalent 187
Boundary coloring, essential 188
Boundary coloring, Kempe sector of a 195
Boundary of a set 49
Boundary point of a set 49
Boundary point of an arc 51
Bounded face 105
Bounded set 44
Bounding circuit of a configuration 157 160 161
Bounding circuit of a graph 194
Bridge 69
Buchroitner, Manfred (* 1950) 11
C-reducible 186 208 214
Caratheodory, Constantin (1873—1950) 29
Cation 223
Cauchy, Augustin Louis (1789—1857) 76
Cayley 13 14 15 128
Chain 145
Chain game see “Kempe chain game”
Chain, 145
Chain, 6-6 229
Chain, closed 152
Chain, disjoint 145
Chain, Kempe 171
Chain, simple 145
Chain, simple closed 152
Charge 223
Chojnacki, later Hanani, Chaim (1912—1991) 27 185 214 221 225 227
Chromodendron 194 199
Cigler, Johann (* 1937) 109
Circuit 69
Circuit edge 69
Circuit, separating 78
Circuit-free 69
Classes of goodness 202
Closed chain 152
Closed Jordan curve 47
Closed set 49
Closed simple—curve 47
Closed unit interval 47
Closed with respect to equivalence 192
COLOR 140 142
Color matrix 193
Color-extendible, directly 192
Color-pair choice 194
Colored graph 171
Coloring, aqimissible 44
Coloring, equivalent 86
Combinatorial graph 141
Compact set 49
Compatible, 210
Compensation 223
Complete (combinatorial) graph 141
Complete 4-gon 101
Complete graph 100
Component 64
Computing time 222
Configuration 155
Configuration, equivalent 159
Configuration, irreducible 168
Configuration, reducible 168
Connected country 45
Connected map 64
Connected, simply 64
continuous 47 50
Contraction 146
Contraction, geometric 149
Contraction, simple 146
Coordinate functions 47
Core of a configuration 156
Counteredge 102
Counterexample, minimal 86
country 62
Country, connected 45
Country, neighboring 80
Cover 195
Coxeter, Harold Scott Macdonald (* 1907) 22
Criminal, minimal 86
Criminal, smallest 86
Cubic map 128
Cundy, H. Martyn 48
Curvature 223
Curve theorem, Jordan 51
Curvilinear triangle 101
D-irreducible 205
D-reducible 186—215
d-vertex 88
Dangling 5-couple 220
de Morgan, Augustus 5 6 10—11 16 104
de Morgan, Sophia (1809—1892) 11
Decomposition 195
Degree of a vertex 88 95
Degree of a vertex (combinatorial) 141
Descartes, Rene (1596—1650) 118
Dingeldey, Friedrich (1859—1939) 22
Dirac, Gabriel Andrews (Gabor) (1925—1984) 24
Discharging procedure 223
Discharging, L 229
Discharging, large 229
Discharging, R 229
Discharging, regular 229
Discharging, S 229
Discharging, small 229
Discharging, T 229
Discharging, transversal 229
Disjoint chains 145
do Carmo, Manftedo P. 45
Domain, exterior, of a circuit 69
Domain, exterior, of a closed chain 153
Domain, exterior, of a closed Jordan curve 51
Domain, exterior, of a ring 153
Domain, interior, of a circuit 69
Domain, interior, of a closed chain 153
Domain, interior, of a closed Jordan curve 51
Domain, interior, of a configuration 156
Domain, interior, of a ring 153
Dual graph 118
Dual map 121
Dual regular solids 120
Dudeney, Henry Ernest 103
Duerre 27—28 30 186 190 192—194 207 222
Duerre number 190
Duerre — Heesch algorithm 187
Dugundji, James (1919—1985) 49—51
EDGE 60
Edge coloring 139
Edge n-coloring 134
Edge, circuit 69
Edge, final, of a map 69
Edge, inner, of a configuration 156
Edge, multiple 99
Edge, outer, of a configuration 156
Eisele, Carolyn 16
Elementary neighborhood 67
Ellis 10 12
Emanating point of a closed curve 52
Embed 160
Embedding, -proper 209
Embedding, proper 194
Empty map 68
End point of an arc 51
Equivalent boundary colorings 187
Equivalent colorings 86
Equivalent configurations 159
Errera, Alfred (1886—1960) 27 185 186
Essential boundary coloring 188
Essential obstruction 220
Euclid (around 300 B.C.) 48
Euler polyhedral formula 117
Euler, Leonhard (1707—1783) 76 118
Exceptional face 194 208
Exterior domain of a circuit 69
Exterior domain of a closed chain 153
Exterior domain of a closed Jordan curve 51
Exterior domain of a ring 153
Face of a graph 105 139
Face, exceptional 208
Face, finite 105
Face, infinite 105
Face, unbounded 105
Fary, Istvin 105 114
Fary, Wagner and, Theorem 111
Federico, P.J. 118
Final edge 69
Final vertex 69
Finite face 105
Five-color theorem 133
Forest 69
Frame 58
France 87
Franklin, Philip (1898—1965) 25 27 184—185 210 221
Fridy, John A. (* 1937) 59
Gelbaum, Bemhard R. 48
Geographically good 228
Geometric contraction 149
Gierloff-Emden, Hans-Guenther (* 1923) 11
Globe 44
Good from the outset 192
Good of stage 1 202
Good of stage n+1 205
Good, 203
Good, geographically 228
Goodness, classes of 202
Graph 61
Graph, colored 171
Graph, combinatorial 141
Graph, complete 100
Graph, complete, (combinatorial) 141
Graph, dual 118
Graph, isomorphic combinatorial 142
Graph, Kuratowski 143
Graph, line 111
Graph, maximal plane 105
Graph, normal 151
Graph, planar 143
Graph, plane 141
Graph, saturated 105
Graph, triangulated 105
Graph, underlying combinatorial 142
Gregory, Duncan Farquharson (1813—1844) 12
Guthrie, Francis 1—2 2
Guthrie, Frederick 5 6 6
Haendler Wolfgang (* 1920) 27
Haken 24 28—30 33—36 36 85 148 183 186 192 215 216 222—223 225 228—230
Hamilton 5 10 21 136
Hanani see “Chojnacki”
Harary, Frank (* 1921) 61
Heawood 22 22 24 25 32 176
Heesch 27—28 28 30 32 118 168 185—186 205 216 219 221 223 227 228
Heesch — Duerre algorithm 187
Heesch's rule of thumb 221
Heffter, Lothar (1862—1962) 118
Heine, Eduard (1821—1881) 59
Hemiite, Charles (1822—1901) 41
Heuser, Harro 48
Hilbert, David (1862—1943) 29 41
hole 44
Homeomorphic 50
Homeomorphism 50
Hopkins, Williams (1793—1866) 12
House of horrors 48
Hub 156
Ibepell, Michael (* 1951) 26
Infinite face 105
Initial point 51
Injective 47
Inner edge of a configuration 156
Inner vertex of a graph 151
Interchange see “Kempe interchange”
Interior domain of a circuit 69
Interior domain of a closed chain 153
Interior domain of a closed Jordan curve 51
Interior domain of a configuration 156
Interior domain of a ring 153
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