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

AbstractA 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 χk(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 χk(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)