## Abstract

The dichromatic number \(\chi (D)\) of a digraph *D*, introduced by Neumann-Lara in the 1980s, is the least integer *k* for which *D* has a coloring with *k* colors such that each vertex receives a color and no directed cycle of *D* is monochromatic. The digraphs considered here are finite and may have antiparalell arcs, but no parallel arcs. A digraph *D* is *k*-critical if each proper subdigraph \(D'\) of *D* satisfies \(\chi (D')<\chi (D)=k\). For integers *k* and *n*, let \(d_k(n)\) denote the minimum number of arcs possible in a *k*-critical digraph of order *n*. It is easy to show that \(d_2(n)=n\) for all \(n\ge 2\), and \(d_3(n)\ge 2n\) for all possible *n*, where equality holds if and only if *n* is odd and \(n\ge 3\). As a main result we prove that \(d_4(n)\ge \lceil (10n-4)/3\rceil\) for all \(n\ge 4\) and \(n\not =5\) where equality holds if \(n \equiv 1 \, \mathrm {(mod} \; 3 \mathrm {)}\) or \(n \equiv 2 \, \mathrm {(mod} \; 3 \mathrm {)}\). As a consequence, we obtain that for each surface \(\mathbf{S}\) the number of 4-critical oriented graphs that can be embedded on \(\mathbf{S}\) is finite.

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Research supported in part by the Simons Visiting Professor, by NSF grant DMS-1600592, and by grants 18-01-00353 and 19-01-00682 of the Russian Foundation for Basic Research.

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Kostochka, A.V., Stiebitz, M. The Minimum Number of Edges in 4-Critical Digraphs of Given Order.
*Graphs and Combinatorics* **36**, 703–718 (2020). https://doi.org/10.1007/s00373-020-02147-y

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DOI: https://doi.org/10.1007/s00373-020-02147-y