On minimal triangle-free graphs with prescribed -defective chromatic number

A graph G is ( m , k ) -colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on vertices receiving the same colour is at most k . The k-defective chromatic number χ k ( G ) is the least positive integer m for which G is ( m , k ) -colourable. Let f ( m , k ) be the smallest order of a triangle-free graph such that χ k ( G ) = m . In this paper we study the problem of determining f ( m , k ) . We show that f ( 3 , 2 ) = 13 and characterize the corresponding minimal graphs. We present a lower bound for f ( m , k ) for all m ≥ 3 and also an upper bound for f ( 3 , k ) .