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A graph is (m, k)-colourable if its vertices can be coloured with m colourssuch that the maximum degree of any subgraph induced on ver- tices receivingthe same colour is at most k. The k-defective chromatic number for a graph isthe least positive integer m for which the graph is (m, k)-colourable. Alltriangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There areexactly four triangle-free graphs of order 9 which have 1-defective chromaticnumber 3. We show that these four graphs appear as subgraphs in almost alltriangle-free graphs of order 10 with 1-defective chromatic number equal to 3.In fact there is a unique triangle-free (3, 1)-critical graph on 10 verticesand we exhibit this graph.

On triangle-free graphs of order 10 with prescribed 1-defective chromatic number