Abstract

The 4-Colour Theorem has been proved in the late seventies [2, 3], after more than a century of fruitless efforts. But the proof has provided very little new information about the map colouring itself. While trying to understand this phenomenon, we analyze colouring in terms of universal properties and adjoint functors. It is well known that the 4-colouring of maps is equivalent to the 3colouring 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. In a sense, this approach seems complementary to the known combinatorial colouring procedures. Current address: Department of Computing, Imperial College, London SW7 2BZ, England; e-mail: D.Pavlovic@doc.ic.ac.uk 1 Introduction: the meaning of t...

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