Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Hat Guessing Number of Graphs   The hat guessing number $HG(G)$ of a graph $G$ on $n$ vertices is defined in
terms of the following game: $n$ players are placed on the $n$ vertices of $G$,
each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible
colors. Each player can see the hat colors of his neighbors, but not his own
hat color. All of the players are asked to guess their own hat colors
simultaneously, according to a predetermined guessing strategy and the hat
colors they see, where no communication between them is allowed. The hat
guessing number $HG(G)$ is the largest integer $q$ such that there exists a
guessing strategy guaranteeing at least one correct guess for any hat
assignment of $q$ possible colors.
  In this note we construct a planar graph $G$ satisfying $HG(G)=12$, settling
a problem raised in \cite{BDFGM}. We also improve the known lower bound of
$(2-o(1))\log_2 n$ for the typical hat guessing number of the random graph
$G=G(n,1/2)$, showing that it is at least $n^{1-o(1)}$ with probability tending
to $1$ as $n$ tends to infinity. Finally, we consider the linear hat guessing
number of complete multipartite graphs.

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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: eess