Spanning trees of bounded degree in random geometric graphs

TL;DR

This paper establishes the precise threshold for embedding all bounded degree trees in random geometric graphs, confirming a conjecture and providing an adaptable, algorithmic proof approach.

Contribution

It determines the sharp threshold for containing all bounded degree trees in random geometric graphs, extending Montgomery's results and confirming a specific conjecture.

Findings

Sharp threshold for bounded degree trees in geometric graphs

Algorithmic proof adaptable to other graph families

Confirmation of a conjecture in the field

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 D\'iaz, Lichev, Mitsche, and Wesolek. Our proof is algorithmic and adapts to other families of graphs, in particular graphs with bounded genus or tree-width.