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 $19$.

翻译：一个有向图的染色是指将其顶点集划分，使得每个独立子集诱导出一个无有向圈的有向图。若有向图$k$-可色，则$k$是这种划分中最小的类数；若每对顶点之间至多存在一条弧，则称该有向图为定向图。显然，最小的$k$-色有向图是$k$个顶点的完全有向图，但确定最小的$k$-色定向图的阶数是一个具有挑战性的问题。已知最小的2-、3-和4-色定向图分别有3、7和11个顶点。1994年，Neumann-Lara猜想最小的5-色定向图有17个顶点。我们解决了这一猜想，并证明正确的阶数是19。