Efficiently four-coloring planar graphs
@inproceedings{Robertson1996EfficientlyFP, title={Efficiently four-coloring planar graphs}, author={Neil Robertson and Daniel P. Sanders and Paul D. Seymour and Robin Thomas}, booktitle={STOC '96}, year={1996} }
An outline of aquadratic algorithm to 4-color planar graphs is presented, based upon anew proof of the Four Color Theorem. This algorithm improves aquartic algorithm of Appel and Haken.
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References
SHOWING 1-10 OF 16 REFERENCES
Every Planar Graph Is 5-Choosable
- MathematicsJ. Comb. Theory, Ser. B
- 1994
We prove the statement of the title, which was conjectured in 1975 by V. G. Vizing and, independently, in 1979 by P. Erdos, A. L. Rubin, and H Taylor.
The Four-Colour Theorem
- MathematicsJ. Comb. Theory, Ser. B
- 1997
Another proof is given, still using a computer, but simpler than Appel and Haken's in several respects, that every loopless planar graph admits a vertex-colouring with at most four different colours.
Every Planar Map Is Four Colorable
- MathematicsMathematical Solitaires & Games
- 2019
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…
Some Simplified NP-Complete Graph Problems
- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1976
Six-Rings in Minimal Five-Color Maps
- Mathematics
- 1947
Introduction. Kempe 1 and Heawood 2 have shown that five colors are sufficient for coloring any map on a sphere. We conjecture the existence of some maps for which five colors are necessary, and seek…