Thoroughly Distributed Colorings
Abstract
We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly distributed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define (G) as the maximum number of colors in such a coloring and (G) as the fractional version thereof. In particular, we show that every claw-free graph with minimum degree at least~2 has~(G) 3/2 and this is best possible. For planar graphs, we show that every triangular disc has (G) 3/2 and this is best possible, and that every planar graph has (G) 4 and this is best possible, while we conjecture that every planar triangulation has (G) 2. Further, although there are arbitrarily large examples of connected, cubic graphs with (G)=1, we show that for a connected cubic graph (G) 2-o(1), and conjecture that it is always at least~2. We also consider the related concepts in hypergraphs.