Strengthened Brooks' Theorem for Digraphs of Girth at least Three

Brooks’ Theorem states that a connected graph G of maximum degree ∆ has chromatic number at most ∆, unless G is an odd cycle or a complete graph. A result of Johansson shows that if G is triangle-free, then the chromatic number drops to O(∆/ log∆). In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph D without directed cycles of length two has chromatic number χ(D) ≤ (1−e−13)∆̃, where ∆̃ is the maximum geometric mean of the out-degree and in-degree of a vertex in D, when ∆̃ is sufficiently large. As a corollary it is proved that there exists an absolute constant α < 1 such that χ(D) ≤ α(∆̃ + 1) for every ∆̃ > 2.