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arXiv · 2006.13294

The Unit Acquisition Number of Binomial Random Graphs

Konstantinos Georgiou·Somnath Kundu·Pawel Pralat
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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.

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Konstantinos Georgiou, Somnath Kundu, Pawel Pralat. 2020-06-23. The Unit Acquisition Number of Binomial Random Graphs. https://arxiv.org/abs/2006.13294

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