Improved Pebbling Bounds

TL;DR

This paper presents new, improved upper bounds on the pebbling number of graphs, enhancing the understanding of pebbling configurations and their solvability.

Contribution

It introduces novel upper bounds on the pebbling number, advancing previous theoretical results in graph pebbling.

Findings

Derived tighter upper bounds on pebbling numbers

Improved theoretical understanding of pebbling configurations

Enhanced bounds applicable to various classes of graphs

Abstract

Consider a configuration of pebbles distributed on the vertices of a connected graph of order $n$. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted $f(G)$, is the minimal number of pebbles such that every configuration of $f(G)$ pebbles on $G$ is solvable. We derive several general upper bounds on the pebbling number, improving previous results.