Acrylic improper colorings of graphs

In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class V i satisses some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most ve and that this result is best possible. For planar graphs, we prove some negative results and state some open problems.