Various bounds on the minimum number of arcs in a k-dicritical digraph
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The dichromatic number χ⃗(G) of a digraph G is the least integer k such that G can be partitioned into k acyclic digraphs. A digraph is k-dicritical if χ⃗(G) = k and each proper subgraph H of G satisfies χ⃗(H) ≤ k-1. cycle of length 2. We prove various bounds on the minimum number of arcs in a k-dicritical digraph, a structural result on k-dicritical digraphs and a result on list-dicolouring. We characterise 3-dicritical digraphs G with (k-1)|V(G)| + 1 arcs. For k ≥ 4, we characterise k-dicritical digraphs G on at least k+1 vertices and with (k-1)|V(G)| + k-3 arcs, generalising a result of Dirac. We prove that, for k ≥ 5, every k-dicritical digraph G has at least (k-1/2 - 1/(k-1)) |V(G)| - k(1/2 - 1/(k-1)) arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree 2(k-1) of a k-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.
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