Abstrakt
In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The 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 eventually linear in t; that is, there are numbers a, b, t_0 such that $pi_t(G)=at+b for all t }= t_0. Our result is also valid for weighted graphs, where every edge e={u,v} has some integer weight \omega(e)}= 2, and a pebbling move from u to v removes \omega(e) pebbles at u and adds one pebble to v.