Geodesic
Vertex Colorings without Rainbow Subgraphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 989-1005

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Given a coloring of the vertices of a graph G, we say a subgraph is rainbow if its vertices receive distinct colors. For a 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 [n/2] + 1.
Keywords: coloring, rainbow, monochromatic, forbidden, path 
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     author = {Goddard, Wayne and Xu, Honghai},
     title = {Vertex {Colorings} without {Rainbow} {Subgraphs}},
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Goddard, Wayne; Xu, Honghai. Vertex Colorings without Rainbow Subgraphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 989-1005. http://geodesic.mathdoc.fr/item/DMGT_2016_36_4_a15/