Thoroughly dispersed colorings

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 td ( G ) as the maximum number of colors in such a coloring and FTD ( G ) as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least  two has  FTD ( G ) ≥ 3 / 2 and this is best possible. For planar graphs, we show that every triangular disc has FTD ( G ) ≥ 3 / 2 and this is best possible, and that every planar graph has td ( G ) ≤ 4 and this is best possible, while we conjecture that every planar triangulation has td ( G ) ≥ 2 . Further, although there are arbitrarily large examples of connected, cubic graphs with td ( G ) = 1 , we show that for a connected cubic graph FTD ( G ) ≥ 2 − o ( 1 ) . We also consider the related concepts in hypergraphs.