t-Pebbling and Extensions
David S. Herscovici, Benjamin D. Hester, Glenn H. Hurlbert

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…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Limits and Structures in Graph Theory · Advanced Graph Theory Research
