The smallest 5-chromatic tournament

Abstract

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 k -chromatic if k 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 k -chromatic digraph is the complete digraph on k k vertices, but determining the order of the smallest k k -chromatic oriented graphs is a challenging problem. It is known that the smallest 2 2 -, 3 3 - and 4 4 -chromatic oriented graphs have 3 3 , 7 7 and 11 11 vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest 5 5 -chromatic oriented graph has 17 17 vertices. We solve this conjecture and show that the correct order is 19 19 .