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{\bf Michael Hoffmann, Ji\v{r}\'{i} Matou\v{s}ek, Yoshio Okamoto and Philipp Zumstein}
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{\bf The $t$-Pebbling Number is Eventually Linear in~$t$}
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In graph pebbling games, one considers a distribution of pebbles
on the vertices of a graph, and a \emph{pebbling move}
consists of taking two pebbles off one vertex and
placing one on an adjacent vertex.
The \emph{$t$-pebbling number} $\pi_t(G)$ of a graph $G$ is the smallest
$m$ such that for every initial distribution of $m$ pebbles
on $V(G)$ and every target vertex $x$ there exists a sequence
of pebbling moves leading to a distibution with at least $t$ pebbles
at $x$. Answering a question of Sieben, we show that for every
graph $G$, $\pi_t(G)$ is
\emph{eventually linear} in $t$; that is, there are
numbers $a,b,t_0$ such that $\pi_t(G)=at+b$ for all $t\ge t_0$.
Our result is also valid for weighted graphs, where every edge
$e=\{u,v\}$ has some integer weight $\omega(e)\ge 2$,
and a pebbling move from $u$ to $v$ removes $\omega(e)$ pebbles
at $u$ and adds one pebble to~$v$.
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