Strengthened Brooks' Theorem for Digraphs of Girth at least Three

Abstract

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 [6] 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 everỹ ∆ > 2.Keywords: Digraph coloring, dichromatic number, Brooks theorem, digon, sparse digraph. * Research supported by FQRNT (Le Fonds québécois de la recherche sur la nature et les technologies) doctoral scholarship. †

Two results on the digraph chromatic number

“…This notion of colorings of digraphs turns out to be the natural way of extending the theory of undirected graph colorings since it provides extensions of most of the basic results from graph coloring theory [3,7,8,14,15].…”

On DP‐coloring of digraphs

“…Regarding digraphs, Erdős [7] conjectured that χ(D) = O( ∆(D) log 2 ∆(D) ) for digon-free digraphs, whereas ∆(D) denotes the maximum total degree of D. To the knowledge of the authors, this conjecture is still open. Related to this question, Harutyunyan and Mohar [9] proved the following. Given a digraph D, let∆(D)…”

The m-degenerate chromatic number of a digraph

“…By taking m = 1 in Theorem 2.10, we obtain Corollary 1.5. Harutyunyan and Mohar [10] posed the question of determining the smallest integer ∆ 0 such that every oriented graph D with ∆ (D) = ∆ 0 satisfies χ A (D) ≤ ∆ 0 − 1. They showed that ∆ 0 exists and is at most some integer which is approximately equal to 10 10 , and they also proved that ∆ 0 ≥ 4.…”

“…Harutyunyan and Mohar [10] applied the probabilistic method to digraphs to show that χ A (D) ≤ (1−e −13 )∆(D) for∆(D) large enough, which only slightly improves upon the trivial bound of χ A (D) ≤ ∆ (D) + 1 and is far from the bound given in Conjecture 1.1. They also posed the following related conjecture, which, although much weaker than Conjecture 1.1 for∆(D) large, gives a precise bound for all∆(D), unlike Conjecture 1.1.…”

“…They also posed the following related conjecture, which, although much weaker than Conjecture 1.1 for∆(D) large, gives a precise bound for all∆(D), unlike Conjecture 1.1. [10]). Let D be an oriented graph.…”