Arc-chromatic number of digraphs in which every vertex has bounded outdegree or bounded indegree

A k-digraph is a digraph in which every vertex has outdegree at most k. A (k ∨ l)digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ(k) (resp. Φ(k, l)) of the arc-chromatic number over the k-digraphs (resp. (k ∨ l)-digraphs). El-Sahili [3] showed that Φ(k, k) ≤ 2k+1. After giving a simple proof of this result, we show some better bounds. We show max{log(2k+3), θ(k+1)} ≤ Φ(k) ≤ θ(2k) and max{log(2k+2l+4), θ(k+1), θ(l+1)} ≤ Φ(k, l) ≤ θ(2k + 2l) where θ is the function defined by θ(k) = min{s : (