The t-Pebbling Number is Eventually Linear in t

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 πt(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, πt(G) is eventually linear in t; that is, there are numbers a, b, t0 such that πt(G) = at + b for all t ≥ t0. Our result is also valid for weighted graphs, where every edge e = {u, v} has some integer weight ω(e) ≥ 2, and a pebbling move from u to v removes ω(e) pebbles at u and adds one pebble to v.