Weighted pebbling numbers on graphs

TL;DR

This paper extends pebbling theory to weighted graphs, providing formulas and examples for weighted pebbling numbers on various graph classes, including stars, trees, and specific paths, highlighting the impact of edge weights.

Contribution

It introduces weighted pebbling numbers, derives explicit formulas for certain graph classes, and demonstrates the necessity of specific edge weights for solvability.

Findings

Weighted pebbling numbers are determined for stars and certain graphs.

Explicit formula for minimum total weight on length-2 paths.

Existence of graphs requiring an edge weight of 1/3 for optimal pebbling.

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 the floor of pw 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 which requires an edge with weight 1/3 in order to achieve its weighted pebbling number.