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Fritsch R., Fritsch G. — The Four-Color Theorem: History, Topological Foundations, and Idea of Proof
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?


Язык: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$d_1 - d_2 - \cdots - d_r$-chain      145
$\Phi$-good      203
$\sigma$-compatible boundary coloring      210
$\sigma$-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, $\Phi$-good      203
Boundary coloring, $\sigma$-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, $d_1 - d_2 - \cdots - d_r$      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, $\sigma$      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, $\sigma$-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, $\Phi$      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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