  The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$
such that $G$ admits a $k$-coloring of its vertex set in such a way that each
color class is an independent set (a set of pairwise non-adjacent vertices).
The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum
$k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that
each color class is acyclic.
  In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most
its circumference, the length of a longest cycle.
  Given a digraph $D$, we will construct three different graphs whose chromatic
numbers bound $\chi_A(D)$.
  Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots,
r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all
cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$,
then $\chi_A(D)\leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$
and $p$, with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length
$r$ modulo $\lceil \frac{k}{p} \rceil p$ for each $r\in
\{-p+2,\ldots,0,\ldots,p\}$, then $\chi_A (D)\leq \lceil \frac{k}{p} \rceil$;
iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference
$c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves,
substantially, the bound proposed by Bondy. Our results show that if we have
more information about the lengths of cycles in a digraph, then we can improve
the bounds for the dichromatic number known until now.
