Abstract
A {\it $k$-digraph} is a digraph in which every vertex has outdegree at most $k$. A {\it $(k\vee l)$-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 $\Phi (k)$ (resp. $\Phi^{\vee}(k,l)$) of the arc-chromatic number over the $k$-digraphs (resp. $(k\vee l)$-digraphs). El-Sahili~\cite{ElS03} showed that $\Phi^{\vee}(k,k)\leq 2k+1$. After giving a simple proof of this result, we show some better bounds. We show $\max\{\log(2k+3), \theta(k+1)\}\leq \Phi(k)\leq \theta(2k)$ and $\max\{\log(2k+2l+4), \theta(k+1), \theta(l+1)\}\leq \Phi^{\vee}(k,l)\leq \theta(2k+2l)$ where $\theta$ is the function defined by $\ds \theta(k)=\min\{s : {s\choose \left\lceil \frac{s}{2}\right\rceil}\geq k\}$. We then study in more details properties of $\Phi$ and $\Phi^{\vee}$. Finally, we give the exact values of $\Phi(k)$ and $\Phi^{\vee}(k,l)$ for $l\leq k\leq 3$.