On defective colourings of triangle-free graphs

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. In 1988 Maddox proved that if either G or Ḡ is triangle-free graph of order p then χk(G)+χk(Ḡ) ≤ 5d p 3k+4e for any integer k ≥ 1. For k = 1, he improved the bound to 6d9e and in 1997 Simanihuruk et.al improved the bound to the sharp upper bound 2 + dp−1 2 e. In this paper we will study the case k = 2 and proved that χ2(G) + χ2(Ḡ) ≤ 2 + d 3 e whenever G is a triangle-free of order p. This improve the upper bound of Maddox for the case k = 2.