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.
Journal of Graph Theory – Wiley
Published: Jan 1, 2018
Keywords: ; ; ; ; ;
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