  A $k$-colouring (not necessarily proper) of vertices of a graph is called
{\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph
induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In
the paper we consider some generalised acyclic $k$-colourings, namely, we
require that each colour class induces an acyclic or bounded degree graph.
Mainly we focus on graphs with maximum degree 5. We prove that any such graph
has an acyclic $5$-colouring such that each colour class induces an acyclic
graph with maximum degree at most 4. We prove that the problem of deciding
whether a graph $G$ has an acyclic 2-colouring in which each colour class
induces a graph with maximum degree at most 3 is NP-complete, even for graphs
with maximum degree 5. We also give a linear-time algorithm for an acyclic
$t$-improper colouring of any graph with maximum degree $d$ assuming that the
number of colors is large enough.
