Weighted Pebbling Numbers of Graphs

Abstract

We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at least one pebble to the target. A pebbling move of $p$ pebbles across an edge with weight $w$ leaves $\lfloor pw\rfloor$ pebbles on the next vertex. We find the weighted pebbling numbers of stars, graphs with at least $2|V|-1$ edges, and trees with given targets. We give an explicit formula for the minimum total weight required on the edges of a length-2 path, solvable with $p$ pebbles, and exhibit a graph that requires an edge with weight $1/3$ in order to achieve its weighted pebbling number.