Acyclic Improper Colorings of Graphs
نویسندگان
چکیده
In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph G is a mapping c from the set of vertices of G to a set of colors such that for every color i, the subgraph induced by the vertices with color i satisses some property depending on i. Such an improper coloring is acyclic if for every two distinct colors i and j, the subgraph induced by all the edges linking a i-colored vertex and a j-colored vertex is acyclic. We prove that every outerplanar graph can be acyclically 2-colored in such a way that every 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.
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