Acyclic T -improper Colourings of Graphs with Bounded Maximum Degree

For graphs of bounded maximum degree, we consider acyclic t-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 t. In the first part, we show that all subcubic graphs are acyclically 1-improperly 3-choosable, thus extending a result of Boiron, Sopena and Vignal (1997, DMTCS 49, 1–10). In the second part, we consider the supremum, over all graphs of maximum degree at most d, of the acyclic t-improper chromatic number and provide t-improper analogues of results by Alon, McDiarmid and Reed (1991, RSA 2(3), 277–288) and Fertin, Raspaud and Reed (2004, JGT 47(3), 163–182). Submission date: 27 February 2007.