Various Bounds on the Minimum Number of Arcs in a k-Dicritical Digraph

The dichromatic number (chi) over right arrow (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 (chi) over right arrow (G) = k and each proper subgraph H of G satisfies (chi) over right arrow (H) S k - 1. 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.