Acyclic improper colourings of graphs with bounded degree

Abstract

In this paper, we continue the study of acyclic improper colourings of graphs introduced in a previous work. An improper colouring of a graph G is a mapping c from the set of vertices of G to a set of colours such that for every colour i, the subgraph induced by the vertices with colour i satisses some property depending on i. Such an improper colouring is acyclic if for every two distinct colours i and j, the subgraph induced by all the edges linking an i-coloured vertex and a j-coloured vertex is acyclic. We consider in this paper the case of graphs with bounded degree. We prove some positive and negative results for graphs with maximum degree three and generalize some of the negative results to graphs with maximum degree k.