  A $(2+1)$-bispindle $B(k_1,k_2;k_3)$ is the union of two $xy$-dipaths of
respective lengths $k_1$ and $k_2$, and one $yx$-dipath of length $k_3$, all
these dipaths being pairwise internally disjoint. Recently, Cohen et al.
conjectured that, for every positive integers $k_1, k_2, k_3$, there is an
integer $g(k_1, k_2, k_3)$ such that every strongly connected digraph not
containing subdivisions of $B(k_1, k_2; k_3)$ has a chromatic number at most
$g(k_1, k_2, k_3)$, and they proved it only for the case where $k_2=1$. For
Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely $g(k_1, k_2,
k_3)\leq 4k$, where $k=max\{k_1, k_2, k_3\}$. A two-blocks cycle $C(k_1,k_2)$
is the union of two internally disjoint $xy$-dipaths of length $k_1$ and $k_2$
respectively. Addario et al. asked if the chromatic number of strong digraphs
not containing subdivisions of a two-blocks cycle $C(k_1,k_2)$ can be bounded
from above by $O(k_1+k_2)$, which remains an open problem. Assuming that
$k=max\{k_1,k_2\}$, the best reached upper bound, found by Kim et al., is
$12k^2$. In this article, we conjecture that this bound can be slightly
improved to $4k^2$ and we confirm our conjecture for some particular cases.
Moreover, we provide a positive answer to Addario et al.'s question for the
class of digraphs having a Hamiltonian directed path.
