Abstract :

The famous planar four colour theorem states that every palanr graph is vertex 4-colourable. While for 1-and 2 colours answer is trivial, one can still ask, which of them need only 3 -colours. Steinberg conjectured that every planar graph that does not contain cycles of length 4 or 5 are -3 colourable. An avelanchee of partial results results showed that planar graphs without cycles of length {4,5,6,...11} , then {4,...,10} etc till {4,..7} are three colourable. We present a sketch of the method used to prove these results with a counterexample to the original conjecture.