On the minimum number of arcs in k-dicritical oriented graphs
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The dichromatic number (D) of a digraph D is the least integer k such that D can be partitioned into k directed acyclic digraphs. A digraph is k-dicritical if (D) = k and each proper subgraph D' of D satisfies (D') ≤ k-1. An oriented graph is a digraph with no directed cycle of length 2. For integers k and n, we denote by o_k(n) the minimum number of edges of a k-critical oriented graph on n vertices (with the convention o_k(n)=+∞ if there is no k-dicritical oriented graph of order n). The main result of this paper is a proof that o_3(n) ≥7n+2/3 together with a construction witnessing that o_3(n) ≤⌈5n/2⌉ for all n ≥ 12. We also give a construction showing that for all sufficiently large n and all k≥ 3, o_k(n) < (2k-3)n, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all k≥ 2, o_k(n) ≥ k - 3/4-1/4k-6 n + 3/4(2k-3), improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.
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