Optimal Pebbling in Products of Graphs

TL;DR

This paper generalizes Graham's Conjecture for optimal pebbling, providing bounds and explicit solutions for various graph products, including complete graphs, cycles, and hypercubes, advancing understanding of pebbling distributions.

Contribution

It extends the theory of optimal pebbling to arbitrary target sets and offers explicit bounds and solutions for complex graph products.

Findings

Bounds on optimal pebbling numbers for products of complete graphs

Explicit optimal t-pebbling numbers for specific graph products

Asymptotic bounds for optimal pebbling numbers of hypercubes

Abstract

We prove a generalization of Graham's Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal $t$-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle $C_5$. Finally, we present explicit distributions which provide asymptotic bounds on optimal pebbling numbers of hypercubes.