Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Islands in minor-closed classes. I. Bounded treewidth and separators   The clustered chromatic number of a graph class is the minimum integer $t$
such that for some $C$ the vertices of every graph in the class can be colored
in $t$ colors so that every monochromatic component has size at most $C$. We
show that the clustered chromatic number of the class of graphs embeddable on a
given surface is four, proving the conjecture of Esperet and Ochem.
Additionally, we study the list version of the concept and characterize the
minor-closed classes of graphs of bounded treewidth with given clustered list
chromatic number. We further strengthen the above results to solve some
extremal problems on bootstrap percolation of minor-closed classes.

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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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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: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph