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, 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 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 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},
}