Lollipop and Cubic Weight Functions for Graph Pebbling

TL;DR

This paper advances graph pebbling theory by developing new weight functions for complex graphs like hypercubes and lollipop graphs, leading to improved bounds and conjectures on pebbling numbers.

Contribution

It introduces a non-tree weight function for $Q_4$, extends weight function techniques to lollipop graphs, and proposes a conjecture for the $n$-dimensional cube.

Findings

Improved upper bounds for pebbling numbers of certain graphs

Construction of valid weight functions for lollipop graph variations

A new conjecture on weight functions for the $n$-cube

Abstract

Given a configuration of pebbles on the vertices of a graph $G$, a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph $G$ is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of $G$ through some sequence of pebbling moves. Through constructing a non-tree weight function for $Q_4$, we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the $n$-dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions.