K. Appel and W. Haken :
“The existence of unavoidable sets of geographically good configurations ,”
Ill. J. Math.
20 : 2
(1976 ),
pp. 218–297 .
MR
392641
Zbl
0322.05141
article
Abstract
People
BibTeX
A set of configurations is unavoidable if every planar map contains at least one element of the set. A configuration \( \mathscr{C} \) is called geographically good if whenever a member country \( M \) of \( \mathscr{C} \) has any three neighbors \( N_1 \) , \( N_2 \) , \( N_3 \) which are not members of \( \mathscr{C} \) then \( N_1 \) , \( N_2 \) , \( N_3 \) are consecutive (in some order) about \( M \) .
The main result is a constructive proof that there exist finite unavoidable sets of geographically good configurations. This result is the first step in an investigation of an approach towards the Four Color Conjecture.
@article {key392641m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The existence of unavoidable sets of
geographically good configurations},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {20},
NUMBER = {2},
YEAR = {1976},
PAGES = {218--297},
URL = {http://projecteuclid.org/euclid.ijm/1256049898},
NOTE = {MR:392641. Zbl:0322.05141.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“Every planar map is four colorable ,”
Bull. Am. Math. Soc.
82 : 5
(September 1976 ),
pp. 711–712 .
MR
424602
Zbl
0331.05106
article
Abstract
People
BibTeX
@article {key424602m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {82},
NUMBER = {5},
MONTH = {September},
YEAR = {1976},
PAGES = {711--712},
DOI = {10.1090/S0002-9904-1976-14122-5},
NOTE = {MR:424602. Zbl:0331.05106.},
ISSN = {0002-9904},
}
K. Appel and W. Haken :
“A proof of the four color theorem ,”
Discrete Math.
16 : 2
(October 1976 ),
pp. 179–180 .
MR
543791
Zbl
0339.05109
article
Abstract
People
BibTeX
@article {key543791m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {A proof of the four color theorem},
JOURNAL = {Discrete Math.},
FJOURNAL = {Discrete Mathematics},
VOLUME = {16},
NUMBER = {2},
MONTH = {October},
YEAR = {1976},
PAGES = {179--180},
DOI = {10.1016/0012-365X(76)90147-3},
NOTE = {MR:543791. Zbl:0339.05109.},
ISSN = {0012-365X},
}
K. Appel and W. Haken :
“Every planar map is four colorable ,”
J. Recreat. Math.
9 : 3
(1976–1977 ),
pp. 161–169 .
MR
543797
Zbl
0357.05043
article
People
BibTeX
@article {key543797m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {Every planar map is four colorable},
JOURNAL = {J. Recreat. Math.},
FJOURNAL = {Journal of Recreational Mathematics},
VOLUME = {9},
NUMBER = {3},
YEAR = {1976--1977},
PAGES = {161--169},
NOTE = {MR:543797. Zbl:0357.05043.},
ISSN = {0022-412x},
}
K. Appel and W. Haken :
“Every planar map is four colorable, I: Discharging ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. 429–490 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977) .
MR
543792
Zbl
0387.05009
article
Abstract
People
BibTeX
We begin by describing, in chronological order, the earlier results which led to the work of this paper. The proof of the Four Color Theorem requires the results of Sections 2 and 3 of this paper and the reducibility results of Part II. Sections 4 and 5 will be devoted to an attempt to explain the difficulties of the Four Color Problem and the unusual nature of the proof.
@article {key543792m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable,
{I}: {D}ischarging},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {429--490},
URL = {http://projecteuclid.org/euclid.ijm/1256049011},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543792. Zbl:0387.05009.},
ISSN = {0019-2082},
}
K. Appel, W. Haken, and J. Koch :
“Every planar map is four colorable, II: Reducibility ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. 491–567 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977) .
MR
543793
Zbl
0387.05010
article
Abstract
People
BibTeX
In Part I of this paper, a discharging procedure is defined which yields the unavoidability (in planar triangulations) of a set \( \mathscr{U} \) of configurations of ring size fourteen or less. In this part, \( \mathscr{U} \) is presented (as Table \( \mathscr{U} \) consisting of Figures 1–63) together with a discussion of the reducibility proofs of its members.
@article {key543793m,
AUTHOR = {Appel, K. and Haken, W. and Koch, J.},
TITLE = {Every planar map is four colorable,
{II}: {R}educibility},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {491--567},
URL = {http://projecteuclid.org/euclid.ijm/1256049012},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543793. Zbl:0387.05010.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“The class check lists corresponding to the supplement to ‘Every planar map is four colorable. Part I and Part II’ ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. C1–C210 .
Microfiche supplement.
Extra material to accompany the supplement published in Ill. J. Math. 21 :3 (1977) .
MR
543794
article
People
BibTeX
@article {key543794m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The class check lists corresponding
to the supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {C1--C210},
NOTE = {Microfiche supplement. Extra material
to accompany the supplement published
in \textit{Ill. J. Math.} \textbf{21}:3
(1977). MR:543794.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“Microfiche supplement to ‘Every planar map is four colorable. Part I and Part II’ ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. 1–251 .
Microfiche supplement.
Supplement to the two part article published as Ill. J. Math. 21 :3 (1977) and Ill. J. Math. 21 :3 (1977) . A class check list was also published as Ill. J. Math. 21 :3 (1977) .
MR
543795
article
People
BibTeX
@article {key543795m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Microfiche supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {1--251},
URL = {https://projecteuclid.org/euclid.ijm/1256049023},
NOTE = {Microfiche supplement. Supplement to
the two part article published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977) and \textit{Ill.
J. Math.} \textbf{21}:3 (1977). A class
check list was also published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977). MR:543795.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“The solution of the four-color-map problem ,”
Sci. Amer.
237 : 4
(October 1977 ),
pp. 108–121 .
MR
543796
article
Abstract
People
BibTeX
@article {key543796m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {The solution of the four-color-map problem},
JOURNAL = {Sci. Amer.},
FJOURNAL = {Scientific American},
VOLUME = {237},
NUMBER = {4},
MONTH = {October},
YEAR = {1977},
PAGES = {108--121},
DOI = {10.1038/scientificamerican1077-108},
URL = {https://www.jstor.org/stable/24953967},
NOTE = {MR:543796.},
ISSN = {0036-8733},
}
K. Appel and W. Haken :
“The four-color problem ,”
pp. 153–180
in
Mathematics today: Twelve informal essays .
Edited by L. A. Steen .
Springer (Berlin ),
1978 .
incollection
People
BibTeX
@incollection {key80470812,
AUTHOR = {K. Appel and W. Haken},
TITLE = {The four-color problem},
BOOKTITLE = {Mathematics today: Twelve informal essays},
EDITOR = {L. A. Steen},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {153--180},
}
K. Appel and W. Haken :
“An unavoidable set of configurations in planar triangulations ,”
J. Comb. Theory, Ser. B
26 : 1
(February 1979 ),
pp. 1–21 .
MR
525813
Zbl
0407.05035
article
Abstract
People
BibTeX
@article {key525813m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {An unavoidable set of configurations
in planar triangulations},
JOURNAL = {J. Comb. Theory, Ser. B},
FJOURNAL = {Journal of Combinatorial Theory. Series
B},
VOLUME = {26},
NUMBER = {1},
MONTH = {February},
YEAR = {1979},
PAGES = {1--21},
DOI = {10.1016/0095-8956(79)90038-8},
NOTE = {MR:525813. Zbl:0407.05035.},
ISSN = {0095-8956},
}
K. Appel, W. Haken, and J. Mayer :
“Triangulation à \( v_5 \) séparés dans le problème des quatre couleurs ”
[Separated triangulation of \( v_5 \) in the four-color problem ],
J. Comb. Theory, Ser. B
27 : 2
(October 1979 ),
pp. 130–150 .
MR
546856
Zbl
0344.05113
article
Abstract
People
BibTeX
Considérant la notion classique minimal planaire 5-chromatique, les auteurs étudient les triangulations du plan dont tons les sommets sont de degré \( \geq 5 \) et dont les sommets de degré 5 sont séparés (aucune arete ne retie deux sommets de degré 5); its prouvent qu’un graphe minimal comporte nécessairement une arête 5-5. L’article présente: 1) une démonstration fondée sur un ensemble minimum de 14 configurations réductibles, 2) une demonstration fondée sur un algorithme applicable au cas général.
@article {key546856m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang and
Mayer, Jean},
TITLE = {Triangulation \`a \$v_5\$ s\'epar\'es
dans le probl\`eme des quatre couleurs
[Separated triangulation of \$v_5\$ in
the four-color problem]},
JOURNAL = {J. Comb. Theory, Ser. B},
FJOURNAL = {Journal of Combinatorial Theory. Series
B},
VOLUME = {27},
NUMBER = {2},
MONTH = {October},
YEAR = {1979},
PAGES = {130--150},
DOI = {10.1016/0095-8956(79)90075-3},
NOTE = {MR:546856. Zbl:0344.05113.},
ISSN = {0095-8956},
}
K. I. Appel :
“Un nouveau type de preuve mathématique. Le théorème des quatre couleurs, II ”
[A new type of mathematical proof: The four-color theorem, II ],
Publ. Dép. Math., Lyon
16 : 3–4
(1979 ),
pp. 81–88 .
In collaboration with W. Haken.
MR
602656
Zbl
0455.05031
article
People
BibTeX
@article {key602656m,
AUTHOR = {Appel, K. I.},
TITLE = {Un nouveau type de preuve math\'ematique.
{L}e th\'eor\`eme des quatre couleurs,
{II} [A new type of mathematical proof:
{T}he four-color theorem, {II}]},
JOURNAL = {Publ. D\'ep. Math., Lyon},
FJOURNAL = {Publications du D\'epartement de Math\'ematiques.
Facult\'e des Sciences de Lyon},
VOLUME = {16},
NUMBER = {3--4},
YEAR = {1979},
PAGES = {81--88},
URL = {http://www.numdam.org/item/PDML_1979__16_3-4_81_0/},
NOTE = {In collaboration with W. Haken. MR:602656.
Zbl:0455.05031.},
ISSN = {0076-1056},
}
K. Appel and W. Haken :
“The four color proof suffices ,”
Math. Intell.
8 : 1
(1986 ),
pp. 10–20 .
MR
823216
Zbl
0578.05022
article
People
BibTeX
@article {key823216m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The four color proof suffices},
JOURNAL = {Math. Intell.},
FJOURNAL = {The Mathematical Intelligencer},
VOLUME = {8},
NUMBER = {1},
YEAR = {1986},
PAGES = {10--20},
DOI = {10.1007/BF03023914},
NOTE = {MR:823216. Zbl:0578.05022.},
ISSN = {0343-6993},
}
K. Appel and W. Haken :
Every planar map is four colorable .
Contemporary Mathematics 98 .
American Mathematical Society (Providence, RI ),
1989 .
With the collaboration of J. Koch.
MR
1025335
Zbl
0681.05027
book
People
BibTeX
@book {key1025335m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {Every planar map is four colorable},
SERIES = {Contemporary Mathematics},
NUMBER = {98},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1989},
PAGES = {xvi+741},
DOI = {10.1090/conm/098},
NOTE = {With the collaboration of J. Koch. MR:1025335.
Zbl:0681.05027.},
ISSN = {0271-4132},
ISBN = {9780821851036},
}