A categorical setting for the 4-Colour Theorem Original Research Article

The 4-Colour Theorem has been proved in the late seventies (Appel and Haken, 1977; Appel et al., 1977), 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 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 3-coloured graphs is cotripleable; every loop-free graph is covered by a 3-coloured one in a universal way. In this context, the 4-Colour Theorem becomes a statement about the existence of a coalgebra structure on graphs. In a sense, this approach seems complementary to the known combinatorial colouring procedures.