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 $\zeta (G)$. 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 $\zeta (G) \le k$ has degeneracy less than $3^k$ and, consequently, satisfies $\chi(G) \le 3^{\zeta (G)}$. 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.

中文翻译：

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本地化数的界限

我们考虑在图形上进行的定位游戏，其中一组警察试图通过利用距离探针来确定一个隐形强盗的确切位置。图 $G$ 的相应优化参数称为定位数，写作 $\zeta (G)$。我们通过提供定位数的上限作为色数的函数来解决 \cite{nisse1} 的猜想。特别地，我们证明了每个具有 $\zeta (G) \le k$ 的图的简并度都小于 $3^k$，因此满足 $\chi(G) \le 3^{\zeta (G)}$ . 我们进一步表明这种简并界是紧密的。我们还证明了外平面图中的定位数最多为 2，并且我们确定了超立方体的定位数，直到一个加性常数。