four colors suffice

1. Four Colors Suffice Dr. Kristine Bauer University of Calgary www.math.ucalgary.ca/~kristine

3. A brief history of the problem 1852 - Guthrie & DeMorgan 1879 - 1880 - first proofs announced (Kempe + Tait) 1890 - 1891 - first proofs renounced (Heawood & Petersen) 1900’s-1970’s - reduction methods. 1976 - Appel & Haken Present - alternative proofs are sought.

4. Graph Theory and 4CT A finite set of vertices.

5. Planar Graphs

6. The Euler Characteristic

7. A proof of the 5 color theorem. Suppose we already know that the 5 color theorem is true for every graph with V-1 or fewer vertices.

8. Constructions with a graph. Consider the graph without v or its edges.

9. Make a new graph by identifying v1 and v2.

10. Algebraic Invariants The Euler characteristic is an algebraic invariant. It can be used on topological spaces as well as graphs. If two spaces have different Euler characteristics, they are different spaces. Spaces which are homeomorphic to spheres have Euler characteristic 2.

11. The five Platonic solids:

12. Proof of the 5 Platonic solids Theorem Each face has n edges, summing up we count every edge twice. Each face has n vertices, summing up we count every vertex k times.Each face has n edges, summing up we count every edge twice. Each face has n vertices, summing up we count every vertex k times.

13. Not all spaces are spheres…

14. Algebraic Topology The field of algebraic topology uses algebraic invariants, like the Euler characteristic, to study spaces. Many of the spaces involved are infinite dimensional. That makes it really hard to study them without some kind of invariant.

15. Further Reading Thomas, Robin “An update on the four color theorem”, Notices of the AMS, vol. 45 no. 7. Devlin, Keith Mathematics: The new golden age, Columbia University Press, 2001. Sato, H. and Hudson, K. Algebraic Topology: an intuitive approach, Translations of Mathematical Monographs, vol. 183, 1996.

16. A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that the figures with any portion of the common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented.

17. Proof of the theorem Reduce to 1476 configurations of regions (using Heesch’s concepts of “reducibility” and “discharging”). More than 1200 hours of computer time on an IBM 360. The first proof of a theorem using a computer. The proof can not be verified by other mathematicians. The proof is now generally accepted. This leaves the reader to face 50 pages containing text and diagrams, 85 pages filled with almost 2500 additional diagrams, and 400 microfiche pages that contain further diagrams and thousands of individual verifications of the claims made in the 24 lemmas in the main sections of text. In addition, the reader is told that certain facts have been verified with the use of about 1200 hours of computer time and would be extremely time-consuming to verify by hand. The papers are somewhat intimidating due to their length and style, and few mathematicians have read them in any detail.This leaves the reader to face 50 pages containing text and diagrams, 85 pages filled with almost 2500 additional diagrams, and 400 microfiche pages that contain further diagrams and thousands of individual verifications of the claims made in the 24 lemmas in the main sections of text. In addition, the reader is told that certain facts have been verified with the use of about 1200 hours of computer time and would be extremely time-consuming to verify by hand. The papers are somewhat intimidating due to their length and style, and few mathematicians have read them in any detail.

18. Seeking a proof without computers. Many equivalent formulations of the problem in combinatorics, algebra and other fields. 1997 - Robertson, Sanders, Seymour, Thomas: Reduced the proof of Appel and Haken to a verifiable computer proof, but still use computer computations. 2004 - Cahit: Claims to have proven the four color theorem using a new concept called spiral chains, and no computers. Has not yet appeared in print.