Strengthened brooks' theorem for digraphs of girth at least three

Abstract

Brooks' Theorem states that a connected graph G of maximum degree Δ has chromatic number at most Δ, 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(Δ/log Δ). 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)̃Δ, wherẽΔ is the maximum geometric mean of the out-degree and in-degree of a vertex in D, wheñΔ is sufficiently large. As a corollary it is proved that there exists an absolute constant α < 1 such that χ (D) ≤ α (̃Δ + 1) for every ̃Δ > 2.