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 Erdős-Rényi 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.
Highlights
We show that asymptotically almost surely au(G(n, m)) = 1 right at the time step the random graph process creates a connected graph
Gossiping and broadcasting are two well studied problems involving information dissemination in a group of individuals connected by a communication network [8]
Before we summarize what is known for the total acquisition number from the perspective of random structures, let us introduce the two models we consider in this paper
Summary
Gossiping and broadcasting are two well studied problems involving information dissemination in a group of individuals connected by a communication network [8]. Examples: In order to warm-up with this graph parameter, note that an acquisition protocol for a cycle C4k (for some k ∈ N) that leaves a residual set of every fourth vertex is the best one can do in any of the three variants of the game; see Figure 1. All legal moves associated with this vertex disconnect the sub-graph induced by vertices with positive weights, implying that all acquisition protocols yield residual sets of size at least 2. Before we summarize what is known for the total acquisition number from the perspective of random structures, let us introduce the two models we consider in this paper. We investigate the unit acquisition of G(n, p) It follows from (1) and the results from [2] that a.a.s. au(G(n, p)) = at(G(n, p)) = 1, provided that p (1+ε) log n/n for some ε > 0.
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