Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Slow-coloring Game on Sparse Graphs: $k$-Degenerate, Planar, and
  Outerplanar   The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$.
Initially, all vertices of $G$ are uncolored. In each round, Lister marks a
nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that
is independent in $G$. The game ends when all vertices are colored. The score
of the game is the sum of the sizes of all sets marked by Lister. The goal of
Painter is to minimize the score, while Lister tries to maximize it. We provide
strategies for Painter on various classes of graphs whose vertices can be
partitioned into a bounded number of sets inducing forests, including
$k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For
example, we show that on an $n$-vertex graph $G$, Painter can keep the score to
at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is
acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual,
$\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$
is planar), and $\frac73n$ when $G$ is outerplanar.

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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: quant-ph