Optimal pebbling number of graphs with given minimum degree

TL;DR

This paper investigates the optimal pebbling number in graphs, improving existing bounds based on minimum degree and diameter, and constructs graphs demonstrating the bounds' sharpness.

Contribution

It refines the upper bounds on the optimal pebbling number for graphs with given minimum degree and diameter, and shows the bounds are tight through infinite graph families.

Findings

Optimal pebbling number is at most 4n/(δ+1), and this bound is sharp.

For graphs with diameter at least 3, the bound improves to 3.75n/(δ+1).

There exist graphs with optimal pebbling number exceeding (8/3 - ε)n/(δ+1).

Abstract

Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $\pi^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{\delta+1}$, where $\delta$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\operatorname{diam}(G)\geq 3$ then we further improve the bound to $\pi^*(G)\leq\frac{3.75n}{\delta+1}$. On the other hand, we show that for arbitrary large diameter and any $\epsilon>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $\left(\frac{8}{3}-\epsilon\right)\frac{n}{(\delta+1)}$.