Spanning trees in random graphs

TL;DR

This paper proves that for any fixed maximum degree, a random graph with a certain edge probability almost surely contains every tree of that size, confirming a conjecture by Kahn.

Contribution

It establishes the existence of spanning trees with bounded degree in random graphs, confirming Kahn's conjecture.

Findings

Random graphs with specified edge probability contain all bounded degree trees

Confirmation of Kahn's conjecture on spanning trees in random graphs

Almost sure presence of all such trees in the specified random graph model

Abstract

For each $\Delta>0$, we prove that there exists some $C=C(\Delta)$ for which the binomial random graph $G(n,C\log n/n)$ almost surely contains a copy of every tree with $n$ vertices and maximum degree at most $\Delta$. In doing so, we confirm a conjecture by Kahn.