Paul Seymour

In 1972, Mader showed that every graph without a 3-connected subgraph is 4-degenerate and thus 5-colourable. We show that the number 5 of colours can be replaced by 4, which is best possible.

We prove that for every path , and every integer , there is a polynomial such that every graph with chromatic number greater than either contains as an induced subgraph, or contains as a subgraph the complete ‐partite graph with parts of cardinality . For and general this is a classical theorem of Gyárfás, and for and general this is a theorem of Bonamy et al.