# The Four-Colour Theorem

@article{Robertson1997TheFT, title={The Four-Colour Theorem}, author={Neil Robertson and Daniel P. Sanders and Paul D. Seymour and Robin Thomas}, journal={J. Comb. Theory, Ser. B}, year={1997}, volume={70}, pages={2-44} }

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken's in several respects.

## 603 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

### Kempe Chains and the Four Colour Problem

- Mathematics
- 1992

In October 1971 the combinatorial world was swept by the rumour that the notorious Four Colour Problem had at last been solved, - that with the help of a computer it had been demonstrated that any…

### A systematic approach to the determination of reducible configurations in the four-color conjecture

- Computer ScienceJ. Comb. Theory, Ser. B
- 1978

### XXIII.—Note on a Theorem in Geometry of Position

- MathematicsTransactions of the Royal Society of Edinburgh
- 1880

In connection with the problem of Map-colouring, I incidentally gave (Proc. K.S.E. 1880, p. 502) a theorem which may be stated as follows : — If 2n points be joined by 3n lines, so that three lines,…

### Une proprie te des graphes minimaux dans le proble me des quatre couleurs, in ``Proble mes Combinatoires et The orie des Graphes,'' Colloques internationaux C.N.R.S

- No. 260,
- 1978

### \Another proof of the four colour theorem { Part I

- Manitoba Conf. on Numerical Math. and Computing, Proc. 7th Congressus Numerantium XX
- 1977