Geodesic
Worm Colorings
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 571-584

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Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a graph F, we define an F-WORM coloring of G as a coloring of the vertices of G without a rainbow or monochromatic subgraph H isomorphic to F. We present some results on this concept especially as regards to the existence, complexity, and optimization within certain graph classes. The focus is on the case that F is the path on three vertices.
Keywords: coloring, rainbow, monochromatic, forbidden, path 
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     author = {Goddard, Wayne and Wash, Kirsti and Xu, Honghai},
     title = {Worm {Colorings}},
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Goddard, Wayne; Wash, Kirsti; Xu, Honghai. Worm Colorings. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 571-584. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a13/