A (vertex-) critical 3-cochromatic graph is either an odd cycle, the complement of an odd cycle, or one of 42 graphs on 6 vertices.
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Research supported by the Danish Natural Science Research Council
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Jørgensen, L.K. Critical 3-cochromatic graphs. Graphs and Combinatorics 11, 263–266 (1995). https://doi.org/10.1007/BF01793013