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Suppose  ℱ = { F  1 , …,  F   t  }  is a collection of distinct subgraphs of a graph   G  = ( V ,  E ) . An    ℱ -WORM   coloring of   G   is the coloring of its vertices such that no copy of each subgraph   F   i   ∈ ℱ  is monochrome or rainbow. This generalizes the notion of   F  -WORM coloring that was introduced recently by W. Goddard, K. Wash, and H. Xu. A (restricted) partition vector  ( ζ   α  , …,  ζ   β  )  is a sequence whose terms   ζ   r    are the number of  ℱ -WORM colorings using exactly   r   colors, with   α  ≤  r  ≤  β  . The partition vectors of complete graphs and those of some  2 -trees are discussed. We show that, although  2 -trees admit the same partition vector in classic proper vertex colorings which forbid monochrome   K  2  , their partition vectors in   K  3  -WORM colorings are different.

ℱ-WORM colorings of some 2-trees: partition vectors