Counterexamples to a Monotonicity Conjecture for the Threshold Pebbling Number

TL;DR

This paper presents counterexamples to a conjecture in graph pebbling, showing that the pebbling number does not always behave monotonically relative to the pebbling threshold, challenging previous assumptions.

Contribution

It provides the first known counterexamples to the monotonicity conjecture for the pebbling number versus the pebbling threshold.

Findings

Counterexamples disprove the conjecture

Pebbling number can be non-monotonic

Challenges previous beliefs in graph pebbling theory

Abstract

Graph pebbling considers the problem of transforming configurations of discrete pebbles to certain target configurations on the vertices of a graph, using the so-called pebbling move. This paper provides counterexamples to a monotonicity conjecture stated by Hurlbert et al. concerning the pebbling number compared to the pebbling threshold.