Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform Bounds for Non-negativity of the Diffusion Game   We study a variant of the chip-firing game called the diffusion game. In the
diffusion game, we begin with some integer labelling of the vertices of a
graph, interpreted as a number of chips on each vertex, and then for each
subsequent step every vertex simultaneously fires a chip to each neighbour with
fewer chips. In general, this could result in negative vertex labels. Long and
Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever
we have a graph on $n$ vertices and an initial allocation with at least $f(n)$
chips on each vertex, then the number of chips on each vertex will remain
non-negative. We answer their question in the affirmative, showing further that
$f(n)=n-2$ is the best possible bound. We also consider the existence of a
similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the
graph.

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The categories of this paper are: adap-org
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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: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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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: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: physics