A Survey of Graph Pebbling

TL;DR

This survey reviews key results and introduces new probabilistic findings on graph pebbling numbers, exploring thresholds in random graphs and their relation to poset properties, thus advancing understanding in graph theory and combinatorics.

Contribution

It provides a comprehensive overview of graph pebbling numbers and presents novel probabilistic thresholds and their connections to poset properties.

Findings

Identified the random graph threshold where pebbling number equals the number of vertices.

Determined pebbling threshold functions for various graph sequences.

Connected pebbling thresholds to the normality of posets and showed the multiset lattice is not supernormal.

Abstract

We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic question of Erd\H os and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal.