Every Planar Map Is Four Colorable

```@inproceedings{Appel1989EveryPM,
title={Every Planar Map Is Four Colorable},
author={Kenneth Appel and Wolfgang Haken},
year={1989}
}```
• Published 1989
• Mathematics
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… Expand
1,220 Citations
Colouring planar graphs
The four-colour problem asks whether every planar graph is 4-colourable and one way to colour planar graphs is to investigate the problem of finding nowhere-zero k-flows in graphs: Theorem 4.1 below asserts that there is an equivalence between k-colouring a planargraph and finding a nowhere- zero k-flow in a graph. Expand
TRIANGULATIONS OF S 3 AND THE COLORING OF GRAPHS
A simple necessary and sufficient condition is given for the vertices of a graph, planar or not, to be properly four-colorable. This criterion involves the notion of an "even" triangulation of S3 andExpand
Extending precolorings of subgraphs of locally planar graphs
• Computer Science, Mathematics
• Eur. J. Comb.
• 2004
The idea of optimal shortcuts is introduced in order to prove the nice cycle lemma and the idea of relative width in orderto prove the main theorem on precolorings with q ≥ 3 colors. Expand
The a-graph coloring problem
This research suggests strongly that the coloring and connectivity conditions for a minimal counterexample are incompatible; infinitely many a-graphs meet one condition or the other, but it is suggested none that meets both. Expand
Light subgraphs of graphs embedded in the plane and in the projective plane – a survey –
• Mathematics
• 2011
It is well known that every planar graph contains a vertex of degree at most 5. A theorem of Kotzig states that every 3-connected planar graph contains an edge whose endvertices have degree-sum atExpand
Coloring Planar graphs via Colored Paths in the associahedra
• Mathematics, Computer Science
• Int. J. Algebra Comput.
• 2013
This paper finds that some recent reformulations of the 4CT are essentially attempting to color elements of ℌ using expressions of elements of F as words in a certain generating set for F, and derives information about not just the colorability of certain elements ofℌ, but also about all possible ways to color these elements. Expand
The a-graph coloring problem: revisiting the 4-color theorem
The existing proofs of the 4-color theorem for planar triangulations do not shed light on why the theorem is true. By examining a related coloring problem that is equivalent to the 4-color problem,Expand
Local girth choosability of planar graphs
• Mathematics
• 2021
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erdős, Rubin, and Taylor in the 1970s. Later, ThomassenExpand
T he four-color theorem [1, 2] is one of the most well-known results in graph theory. Originating from the question of coloring a world map, posed in the middle of the nineteenth century, it hasExpand
Three Colors Suffice: Conflict-Free Coloring of Planar Graphs
The conflict-free variant of the famous Hadwiger Conjecture is proved: If G does not contain Kk+1 as a minor, then χCF(G) ≤ k, and a tight worst-case bound is obtained: three colors are sometimes necessary and always sufficient for planar graphs. Expand

References

SHOWING 1-7 OF 7 REFERENCES
A systematic approach to the determination of reducible configurations in the four-color conjecture
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1978
A comprehensive list of reducible configurations up to the 10-ring level is tabled and the effectiveness of the program is improved after implementing results obtained by investigating the algebraic structure of the problem. Expand
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
then high school students and undergraduates) for aiding in checking the diagrams in the preprints for the papers