Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Distinguishability of infinite groups and graphs   The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$
is the least number of colors needed to color the elements of $V$ so that no
non-identity element of the group preserves the coloring. The {\em
distinguishing number} of a graph is the distinguishing number of its full
automorphism group acting on its vertex set. A connected graph $\Gamma$ is said
to have {\em connectivity 1} if there exists a vertex $\alpha \in V\Gamma$ such
that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an
orbit of the point stabilizer $G_\alpha$ is called a {\em suborbit} of $G$.
  We prove that every connected primitive graph with infinite diameter and
countably many vertices has distinguishing number 2. Consequently, any
infinite, connected, primitive, locally finite graph is 2-distinguishable; so,
too, is any infinite primitive group with finite suborbits. We also show that
all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian
products of connected denumerable graphs of infinite diameter have
distinguishing number 2. All of our results follow directly from a versatile
lemma which we call The Distinct Spheres Lemma.

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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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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: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg