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arXiv · 1804.03717

Optimal pebbling number of graphs with given minimum degree

Andrzej Czygrinow·Glenn Hurlbert·Gyula Y. Katona·László F. Papp
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Abstract

Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $\pi^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{\delta+1}$, where $\delta$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\operatorname{diam}(G)\geq 3$ then we further improve the bound to $\pi^*(G)\leq\frac{3.75n}{\delta+1}$. On the other hand, we show that for arbitrary large diameter and any $\epsilon>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $\left(\frac{8}{3}-\epsilon\right)\frac{n}{(\delta+1)}$.

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Andrzej Czygrinow, Glenn Hurlbert, Gyula Y. Katona, László F. Papp. 2018-04-10. Optimal pebbling number of graphs with given minimum degree. https://arxiv.org/abs/1804.03717

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