Digraph Girth via Chromatic Number

Let D be a digraph. The chromatic number χ(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or ∞ if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with χ(D) > k. It is shown that G(k, n) ≥ n1/k and G(k, n) ≤ (3 log2 n log2 log2 n)1−1/kn1/k for n ≥ 3 and k ≥ 2.