An Open Problem on k-Defective Colourings of Triangle-free Graphs and their Complements

A graph G 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-defective chromatic number χk(G) is the least positive integer m for which G is (m,k)colourable. Maddox proved that χk(G)+χk(Ḡ) ≤ 5 p 3k+4 whenever G is a triangle-free of order p and k ≥ 0 is an integer. Simanihuruk et al proved that Maddox’s upper bound is a weak upper bound for k = 1. In this paper a better upper bound of χk(G) + χk(Ḡ) is established whenever G is a triangle-free graph and k = 2. Hence finding a sharp upper bound of χk(G) + χk(Ḡ) is an open problem whenever G is a triangle-free graph. Mathematics Subject Classification: 05C15, 05C35