Every Planar Map Is Four Colorable

@article{Appel2019EveryPM,
  title={Every Planar Map Is Four Colorable},
  author={Kenneth Appel and Wolfgang Haken},
  journal={Mathematical Solitaires \& Games},
  year={2019}
}
  • K. Appel, W. Haken
  • Published 19 March 2019
  • Mathematics
  • Mathematical Solitaires & Games
As has become standard, the four color map problem will be considered in the dual sense as the problem of whether the vertices of every planar graph (without loops) can be colored with at most four colors in such a way that no pair of vertices which lie on a common edge have the same color. The restriction to triangulations with all vertices of degree at least five is a consequence of the work of A. B. Kempe. Over the past 100 years, a number of authors including A. B. Kempe, G. D. Birkhoff… 
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References

SHOWING 1-8 OF 8 REFERENCES
The existence of unavoidable sets of geographically good configurations
A set of configurations is unavoidable if every planar map contains at least one element of the set. A configuration r is called geographically good if whenever a member country M of c has any three
Questions raised by Dan Younger in 1987 helped us to provide an appropriate level of detail in the sections on immersion reducibility in the appendix to Part II
  • 1987
Bibliographisches Institut
  • Bibliographisches Institut
  • 1969
Kempe chains and the four color problem, Utilitas Math
  • MR 46 #8887. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS
  • 1972
then high school students and undergraduates) for aiding in checking the diagrams in the preprints for the papers
  • In addition,
  • 1977