The Unit Acquisition Number of Binomial Random Graphs

TL;DR

This paper studies the unit acquisition number in Erdős-Rényi random graphs, showing that it almost surely becomes 1 exactly when the graph becomes connected, revealing a sharp phase transition.

Contribution

It proves that the unit acquisition number of Erdős-Rényi graphs drops to 1 precisely at the connectivity threshold, establishing a strong link between graph connectivity and acquisition number.

Findings

a_u(G) = 1 at the connectivity threshold

The acquisition number is at least 2 if the graph is disconnected

The result holds asymptotically almost surely

Abstract

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The unit acquisition number of $G$, denoted by $a_u(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erd\H{o}s-R\'{e}nyi random graph process $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choose 2}$. We show that asymptotically almost surely $a_u(\mathcal{G}(n,m)) = 1$ right at the time step the random graph process creates a connected graph. Since trivially $a_u(\mathcal{G}(n,m)) \ge 2$ if the graphs is disconnected, the result holds in the strongest possible sense.