完全分布的着色
Thoroughly Distributed Colorings
摘要：
我们考虑（不一定是正确的）着色的顶点的图，每一个颜色是完全分布的，也就是说，出现在每个开放的邻域。等价地，每个颜色都是一个全控制集。我们将$\td(G)$定义为这种着色中的最大颜色数，而将$\FTD(G)$定义为其分数版本。特别地，我们证明了每个最小度至少为~ $2$的无爪图都有~ $\FTD(G)\ge 3/2$，并且这是最好的可能。对于可平面图，我们证明了每个三角形圆盘都有$\FTD(G) \ge 3/2$，这是最好的可能，每个平面图都有$\td(G) \le 4$，这是最好的可能，而我们猜想每个平面三角剖分有$\td(G)\ge 2$。此外，虽然有任意大的例子，连接，三次图与$\td(G)=1$，我们表明，对于一个连接的三次图$\FTD(G) \ge 2-o(1)$，并推测，它总是至少~ $2$。我们还考虑了超图中的相关概念。
英文摘要：
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.
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