Bjarne Toft

An Abstract Generalization of a Map Reduction Theorem of Birkhoff

In 1913, George D, Birkhoff proved several theorems for planar maps, reducing the 4-colourability of maps containing certain configurations to the 4-colourability of smaller maps. One such result was that rings of size at most 4 are reducible. This was generalized by G. A. Dirac in 1960 to the abstract formulation that any contraction-critical k-chromatic graph ? Kk is 5-connected. In the same spirit we generalize the reducibility of a 6-ring around 4 countries, each having 5 neighbours (sometimes called Birkhoff?s diamond theorem) to the statement that in a contraction-critical k-chromatic graph ? Kk no four vertices of degree k span a complete 4-graph with a missing edge. This is subsequently used to prove that the number of vertices of degree ? k + 1 must be at least k ? 4. It is remarked that such a result for all k with k ? 4 replaced by ck ? d, where c > 1, would imply Hadwiger?s conjecture that there are no contraction-critical k-chromatic graphs ? Kk for all k.

14. Chromatic Polynomials

If one were to delve into the fascinating combinatorial branch of mathematics that is graph theory, it would not take too long at all for one to encounter the infamous four colour theorem. It was proposed that for any planar graph, it would always be possible to colour any vertex in one of four colours such that no pair of vertices connected by a shared edge would be of the same colour. This theorem, formally proposed in 1852, had been left unproven for years, baffling the minds of great mathematicians for over a century, until it was rigorously proven in 1976 by Kenneth Appel and Wolfgang Haken, with the aid of computer calculation and algorithms which were obviously not to hand at the time [2]. In the decades that the proof was left undiscovered, many mathematicians have tried to tackle the problem with various approaches. One attempt proposed by George David Birkhoff in 1912, was the introduction of chromatic polynomials, a set of polynomials in variable λ which could be used to define the number of possible colourings of a given graph using λ colours. If Birkhoff could prove that every chromatic polynomial of a planar graph was strictly greater than zero in the case where λ = 4, then each planar graph would have at least one 4-colouring, which would prove the four colour theorem to be correct [1]. Sadly, Birkhoffs efforts were unfruitful for their initial goal. However, chromatic polynomials have been since further studied, with mathematicians achieving results that will be shown in this very paper, which have been helpful in finding the number of colourings for various graphs, compounds and augmentations.