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arXiv · 2303.14229

Sharp threshold for embedding balanced spanning trees in random geometric graphs

Alberto Espuny Díaz·Lyuben Lichev·Dieter Mitsche·Alexandra Wesolek
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Abstract

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $\mathcal{G}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

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Alberto Espuny Díaz, Lyuben Lichev, Dieter Mitsche, Alexandra Wesolek. 2023-03-24. Sharp threshold for embedding balanced spanning trees in random geometric graphs. https://arxiv.org/abs/2303.14229

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