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{\bf Mark E. Watkins and Xiangqian Zhou}
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{\bf Distinguishability of Locally Finite Trees}
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The {\em distinguishing number} $\Delta(X)$ of a graph $X$ is the
least positive integer $n$ for which there exists a function
$f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of
$\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$,
$k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees
without pendant vertices are shown to be 2-distinguishable. A proof
is indicated that extends 2-distinguishability to locally countable
trees without pendant vertices. It is shown that every infinite,
locally finite tree $T$ with finite distinguishing number contains a
finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results
are obtained for the {\em distinguishing chromatic number}, namely the
least positive integer $n$ such that the function $f$ is also a proper
vertex-coloring.
\bye