2022/10/18 by Thomas Bellitto, Bellitto, Thomas, Nicolás Bousquet +5 · 1 citation
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2210.09936
openalex publication_date 2022/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.