The Nordhaus-Gaddum problem for the k-defective chromatic number of a P4-free graph

Abstract

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-defective chromatic number Xk(G) of a graph G is the least positive integer m for which G is (m, k)-colourable. The Nordhaus-Gaddum problem is to find sharp bounds for Xk(G)+Xk(G) and Xk(G). Xk(G) over the set of all graphs of order p where G is the complement of the graph G. In this paper we obtain a sharp upper bound for Xk(G)+Xk(G), where G is a P4-free graph of order p and k = 1 or 2.