# Total Aquisition Number of Random Geometric Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the total weight of a vertex $u$ can be moved to a neighbouring vertex $v$, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process.

We will investigate the case where $G=G(n,r)$ is a random geometric graph in which $n$ vertices are distributed u.a.r. on $[0,\sqrt{n}]^2$, and two vertices are adjacent iff the distance between them is at most $r$. We prove that a.a.s. $a_t(G(n,r)) = \Theta( n / (r \lg r)^2)$ if $r < 1$, and $a_t(G(n,r)) = \Theta(1)$ if $r \lg r > \sqrt{n}$.

This is joint worl with D. Mitsche and P. Pralat.