本地化编号的界限
Bounds on the localization number
摘要：
我们考虑的本地化游戏上的图，其中一组警察试图确定一个无形的强盗的确切位置，利用距离探针。图$G$的相应优化参数称为局部化数，并写为$ζ(G)$。我们解决了一个猜想的\cite{nisse1}的本地化数的色数的函数提供了一个上界。特别是，我们证明了每一个图与$ζ(G) \le k$退化小于$3^k$，因此，满足$χ(G) \le 3^{ζ(G)}$。我们进一步表明，这个简并界是紧的。我们还证明了在外平面图中的局部化数至多为2，并且我们确定了超立方体的局部化数，直到一个加性常数。
英文摘要：
We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph $G$ is called the localization number and is written $ζ(G)$. We settle a conjecture of \cite{nisse1} by providing an upper bound on the localization number as a function of the chromatic number. In particular, we show that every graph with $ζ(G) \le k$ has degeneracy less than $3^k$ and, consequently, satisfies $χ(G) \le 3^{ζ(G)}$. We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.
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