The content discusses the minimum number of arcs in 4-dicritical oriented graphs. A digraph D is k-dicritical if its dichromatic number ⃗χ(D) = k and each proper subdigraph H of D satisfies ⃗χ(H) < k.
The authors prove that every 4-dicritical oriented graph on n vertices has at least (10/3 + 1/51)n - 1 arcs. This shows that the conjecture by Kostochka and Stiebitz, which states that for any k ≥ 3, there is a constant αk > 0 such that ok(n) > (1 + αk)dk(n) for n sufficiently large, holds for k = 4.
The authors also characterize exactly the 4-dicritical digraphs on n vertices with exactly 10/3 n - 4/3 arcs, which are called the 4-Ore digraphs. The proof uses the potential method and involves constructing smaller 4-dicritical digraphs by identifying vertices of the original digraph.
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by Fréd... às arxiv.org 04-30-2024
https://arxiv.org/pdf/2306.10784.pdfPerguntas Mais Profundas