A note describing A CATEGORICAL SETTING FOR THE 4-COLOUR THEOREM by Dusko Pavlovic is available by anonymous ftp. Abstract: It is well known that the 4-colouring of maps is equivalent to the 3-colouring of the edges of some graphs. We show that every slice of the category of 3-coloured graphs is a topos. The forgetful functor to the category of graphs is cotripleable; every loop-free graph is covered by a 3-coloured one in a universal way. In this context, the 4-Color Theorem becomes a statement about the existence of coalgebra structure on graphs. The "projective" approach to graphs, described here, is, in a sense, dual to the usual combinatorial treatment, based on induction. I shall try to relate the two approaches in another paper. How to get a copy:
ftp triples.math.mcgill.ca login: anonymous password:[your e-mail address] cd pub/pavlovic bin get 4color-US.ps.Z %if your printer has American standards %or get 4color-A4.ps.Z %otherwise bye uncompress 4color-++.ps.Z
If you have any problems printing out this PS-file, please let me know. (I am not distributing the DVI or LaTeX versions of the paper because it contains several PS-diagrams.) Regards, Dusko ==============================================================================
Please also note "A categorical characterization of the four colour theorem"by Barry Fawcett, Canad. Math. Bull. Vol. 29 (4), 1986, perhaps the main result of which is the equivalence of the four colour theorem with the statement that epimorphisms are surjective in the category of planar graphs. Wood ==============================================================================
Thanks, Richard. As you could see if you FTP-ed my note, I did quote Barry Fawcett's characterisation (and thanked Robert Pare for mentioning it to me). I also explained that Fawcett's approach is, in a sense, dual to mine. I hope I'll be able to say more about this in another note. Best regards, Dusko Pavlovic ==============================================================================
participants (2)
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pavlovic@triples.Math.McGill.CA -
Richard Wood