Spanning trees of bounded degree in random geometric graphs
Michael Anastos, Sahar Diskin, Dawid Ignasiak, Lyuben Lichev, Yetong Sha
TL;DR
This paper addresses the problem of universality for containing all $n$-vertex trees with bounded maximum degree in the random geometric graph $G_d(n,r)$. It identifies a sharp threshold location $r_c(n,d,\Delta)= \frac{\sqrt{d}\log(\Delta-1)}{2\log n}$ and proves universality for $r \ge (1+\varepsilon) r_c$ while non-universality holds for $r \le (1-\varepsilon) r_c$ for suitable $\varepsilon$ (e.g., $\varepsilon = \frac{100 d\log (\Delta \log n)}{\log n}$). The method is algorithmic, relying on a tree-decomposition lemma to partition trees into subtrees and a controlled embedding in a tessellated unit cube with anchors and layered structures, complemented by probabilistic concentration to guarantee point availability. The results extend Montgomery’s binomial-threshold universality to the geometric setting and adapt to graphs with bounded genus or tree-width, contributing a geometry-aware universality framework and raising open questions on hitting times and almost-all-tree thresholds.
Abstract
We determine the sharp threshold for the containment of all $n$-vertex trees of bounded degree in random geometric graphs with $n$ vertices. This provides a geometric counterpart of Montgomery's threshold result for binomial random graphs, and confirms a conjecture of Espuny Díaz, Lichev, Mitsche, and Wesolek. Our proof is algorithmic and adapts to other families of graphs, in particular graphs with bounded genus or tree-width.