Modified Linear Programming and Class 0 Bounds for Graph Pebbling

TL;DR

This paper improves bounds on the pebbling number of graphs using modified linear programming techniques and explores structural properties of Class 0 graphs, disproving a previous conjecture and establishing tight bounds for diameter 2 graphs.

Contribution

It introduces a new set of weight functions for linear programming in graph pebbling and provides new bounds on the number of edges in Class 0 graphs, including disproving a conjecture.

Findings

Improved bounds on pebbling numbers for certain graphs.

Disproved a conjecture regarding edges in Class 0 graphs.

Established tight bounds for diameter 2 graphs with no cut-vertex.

Abstract

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.