Critical Pebbling Numbers of Graphs

Abstract

We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-\emph{critical pebbling numbers}. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields $7$ graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.