Multitrees in random graphs

TL;DR

This paper investigates the emergence of MultiTrees in random graphs, establishing thresholds for their existence and providing bounds for different numbers of spanning trees.

Contribution

It introduces the concept of MultiTrees in random graphs and determines the minimum size needed for their likely appearance.

Findings

Hitting time result for s=2 MultiTrees

O(n log n) bound for s≥3 MultiTrees

Thresholds for MultiTree existence in random graphs

Abstract

Let $N=\binom{n}{2}$ and $s\geq 2$. Let $e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$ be $s$ independent permutations of the edges $E(K_n)$ of the complete graph $K_n$. A {\em MultiTree} is a set $I\subseteq [N]$ such that the edge sets $E_{I,j}$ induce spanning trees for $j=1,2,\ldots,s$. In this paper we study the following question: what is the smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a hitting time result for $s=2$ and an $O(n\log n)$ bound for $s\geq 3$.