On defective colourings of complementary graphs

A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on vertices receiving the same colour is at most k. The k-defedive chromatic number Xk(G) of a graph G is the least positive integer m for which G is (m,k)-colourable. In this paper we obtain a sharp upper bound for XI (G) + X/G) whenever G has no induced subgraph isomorphic to P 4 ' a path of order four. For general k, we obtain a weak upper bound for Xk(G) + Xk(G). Furthermore we will present a sharp lower bound for the product Xk(G).Xk(G) in terms of some generalized Ramsey numbers and discuss the associated realizability problem for the I-defective chromatic number.