On the intersection of pairs of trees

TL;DR

This paper studies the distribution of common edges in pairs of random spanning trees across different graph types, proving Poisson limit laws for complete graphs, Erdős–Rényi graphs, and multipartite graphs.

Contribution

It provides new proofs and extends Poisson limit results for the number of shared edges in pairs of random spanning trees to various graph classes.

Findings

Poisson distribution with mean 2 for complete graphs

Poisson limit law for Erdős–Rényi graphs with constant p

Extension of results to complete multipartite graphs

Abstract

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson distribution with expected value $2$. This is applied to show a Poisson limit law for the number of common edges in two independent random spanning trees of an Erd\H{o}s--R\'enyi random graph $G(n,p)$ for constant~$p$. We also use the same method to prove an analogous result for complete multipartite graphs.