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arXiv · 2310.00580

Lollipop and Cubic Weight Functions for Graph Pebbling

Marshall Yang·Carl Yerger·Runtian Zhou
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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.

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Marshall Yang, Carl Yerger, Runtian Zhou. 2024-11-24. Lollipop and Cubic Weight Functions for Graph Pebbling. https://arxiv.org/abs/2310.00580

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