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- W2528174379 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 $td(G)$ 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~$2$ has~$FTD(G)ge 3/2$ and this is best possible. For planar graphs, we show that every triangular disc has $FTD(G) ge 3/2$ and this is best possible, and that every planar graph has $td(G) le 4$ and this is best possible, while we conjecture that every planar triangulation has $td(G)ge 2$. Further, although there are arbitrarily large examples of connected, cubic graphs with $td(G)=1$, we show that for a connected cubic graph $FTD(G) ge 2-o(1)$, and conjecture that it is always at least~$2$. We also consider the related concepts in hypergraphs." @default.
- W2528174379 created "2016-10-14" @default.
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- W2528174379 date "2016-09-30" @default.
- W2528174379 modified "2023-09-27" @default.
- W2528174379 title "Thoroughly Distributed Colorings" @default.
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