The smallest 5-chromatic tournament
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A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is k-chromatic if k is the minimum number of classes in such partition, and a digraph is oriented if there is at most one arc between each pair of vertices. Clearly, the smallest k-chromatic digraph is the complete digraph on k vertices, but determining the order of the smallest k-chromatic oriented graphs is a challenging problem. It is known that the smallest 2-, 3- and 4-chromatic oriented graphs have 3, 7 and 11 vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest 5-chromatic oriented graph has 17 vertices. We solve this conjecture and show that the correct order is 19.
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