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.