  A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily
proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and
$(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph
homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate
the study of the computational complexity of tropical graph homomorphism
problems. We consider two settings. First, when the tropical graph $(H,c)$ is
fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring
of $H$ is part of the input; the associated decision problem is called
$H$-TROPICAL-COLOURING. Each $(H,c)$-COLOURING problem is a constraint
satisfaction problem (CSP), and we show that a complexity dichotomy for the
class of $(H,c)$-COLOURING problems holds if and only if the Feder-Vardi
Dichotomy Conjecture for CSPs is true. This implies that $(H,c)$-COLOURING
problems form a rich class of decision problems. On the other hand, we were
successful in classifying the complexity of at least certain classes of
$H$-TROPICAL-COLOURING problems.
