Acyclic improper colourings of graphs with bounded maximum degree

For graphs of bounded maximum degree, we consider acyclic tt-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic, and each colour class induces a graph with maximum degree at most tt.We consider the supremum, over all graphs of maximum degree at most dd, of the acyclic tt-improper chromatic number and provide tt-improper analogues of results by Alon, McDiarmid and Reed [N. Alon, C.J.H. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Structures Algorithms 2 (3) (1991) 277–288] and Fertin, Raspaud and Reed [G. Fertin, A. Raspaud, B. Reed, Star coloring of graphs, J. Graph Theory 47 (3) (2004) 163–182].