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

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