Vertex colorings without rainbow subgraphs

Given a coloring of the vertices of a graph G, we say a subgraph is rainbow if its vertices receive distinct colors. For graph F , we define the F -upper chromatic number of G as the maximum number of colors that can be used to color the vertices of G such that there is no rainbow copy of F . We present some results on this parameter for certain graph classes. The focus is on the case that F is a star or triangle. For example, we show that the K3-upper chromatic number of any maximal outerplanar graph on n vertices is bn/2c+ 1.