The hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erdős for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than 2¹²⁵⁰⁰⁰. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered planar graph has bounded hat guessing number.

中文翻译：

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关于平面图类的帽子猜测数

猜帽数是基于 Winkler 引入的猜帽游戏的图不变量。使用涉及超图的 Erdős 边密度定理的新顶点分解参数，我们表明所有外平面图的帽子猜测数小于 2 ¹²⁵⁰⁰⁰。我们还定义了包含外平面图的分层平面图类，并且我们证明了每个分层平面图都有有界的帽子猜测数。