t-Pebbling and Extensions

TL;DR

This paper advances graph pebbling theory by establishing bounds on t-pebbling numbers for diameter two graphs, providing efficient algorithms, and exploring fractional pebbling variants, with implications for various graph classes.

Contribution

It proves a conjecture on t-pebbling bounds for diameter two graphs, introduces efficient algorithms, and investigates fractional pebbling numbers, extending the understanding of pebbling in different graph types.

Findings

Established the best upper bound on t-pebbling number for diameter two graphs.

Developed a linear time algorithm for t-pebbling in diameter two graphs.

Proved the fractional pebbling number conjecture of Hurlbert.

Abstract

Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll). For t >= 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex. We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis, et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack. Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs. Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.