Minimum number of arcs in k-critical digraphs with order at most 2k-1

The dichromatic number χ⃗(D) of a digraph D is the least integer k for which D has a coloring with k colors such that there is no monochromatic directed cycle in D. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph D is called k-critical if each proper subdigraph D' of D satisfies χ⃗(D')<χ⃗(D)=k. For integers k and n, let ext(k,n) denote the minimum number of arcs possible in a k-critical digraph of order n. It is easy to show that ext(2,n)=n for all n≥ 2, and ext(3,n)≥ 2n for all possible n, where equality holds if and only if n is odd and n≥ 3. As a main result we prove that if n, k and p are integers with n=k+p and 2≤ p ≤ k-1, then ext(k,n)=2(n2 - (p^2+1)), and we give an exact characterisation of k-critical digraphs for which equality holds. This generalizes a result about critical graphs obtained in 1963 by Tibor Gallai.