Pebbling on Directed Graphs with Fixed Diameter

Abstract

Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $\pi(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $\pi(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected graph on $n$ vertices with diameter 2 has $\pi(G) = n$ unless it belongs to an exceptional family of graphs, consisting of those that can be constructed in a specific manner; in which case $\pi(G) = n +1$. By generalizing a result of Chan and Godbole, Postle showed that for a graph with diameter $d$, $\pi(G) \le n 2^{\lceil \frac{d}{2} \rceil} (1+o_n(1))$. In this article, we continue this study relating pebbling and diameter with a focus on directed graphs. This leads to some surprising results. First, we show that in an oriented directed graph $G$ (in the sense that if $i \to j$ then we cannot have $j \to i$), it is indeed the case that if $G$ has diameter 2, $\pi(G) = n$ or $n + 1$, and if $\pi(G) = n+1$, the directed graph has a very particular structure. In the case of general directed graphs (that is, if $i \to j$, we may or may not have an arc $j \to i$) with diameter 2, we show that $\pi(G)$ can be as large as $\frac32 n + 1$, and further, this bound is sharp. More generally, we show that for general directed graphs, $\pi(G) \le 2^d n / d + f(d)$ where $f(d)$ is some function of only $d$.