Random subtrees of complete graphs

TL;DR

This paper investigates the asymptotic probabilities and expectations related to subtrees of complete graphs, revealing that certain probabilities converge to approximately 0.692 and the expected subtree size approaches n-1.

Contribution

It provides new asymptotic results for probabilities and expectations of subtrees in complete graphs, including convergence values for these statistics.

Findings

Both probabilities approach approximately 0.692.

Expected subtree size approaches n-1.

Random subtrees are typically near spanning trees in size.

Abstract

We study the asymptotic behavior of four statistics associated with subtrees of complete graphs: the uniform probability $p_n$ that a random subtree is a spanning tree of $K_n$, the weighted probability $q_n$ (where the probability a subtree is chosen is proportional to the number of edges in the subtree) that a random subtree spans and the two expectations associated with these two probabilities. We find $p_n$ and $q_n$ both approach $e^{-e^{-1}}\approx .692$, while both expectations approach the size of a spanning tree, i.e., a random subtree of $K_n$ has approximately $n-1$ edges.