| Abstract: | We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly dispersed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define  as the maximum number of colors in such a coloring and  as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least two has  and this is best possible. For planar graphs, we show that every triangular disc has  and this is best possible, and that every planar graph has  and this is best possible, while we conjecture that every planar triangulation has  . Further, although there are arbitrarily large examples of connected, cubic graphs with  , we show that for a connected cubic graph  . We also consider the related concepts in hypergraphs. |
