The smallest 5-chromatic tournament

TL;DR

This paper resolves a longstanding conjecture by proving that the smallest 5-chromatic oriented graph has 19 vertices, refining previous bounds and advancing understanding of digraph colorings.

Contribution

The paper proves that the minimal order of a 5-chromatic oriented graph is 19, confirming and correcting Neumann-Lara's 1994 conjecture.

Findings

The smallest 5-chromatic oriented graph has 19 vertices.

Previous bounds for 2-, 3-, 4-, and 5-chromatic graphs are refined.

The conjecture by Neumann-Lara is confirmed and corrected.

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 $k$-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 $19$.