Embedding spanning trees in random graphs

TL;DR

This paper establishes conditions under which a random graph G(n,p) almost surely contains any spanning tree T with bounded maximum degree, based on the edge probability p(n).

Contribution

It provides a nearly tight bound on the edge probability needed for embedding spanning trees with bounded degree in random graphs.

Findings

The probability p(n) must satisfy np>c*max{D*logn,n^{ heta}} for embedding.

Embedding is almost surely possible under these conditions for large n.

The bounds are tight for trees with maximum degree proportional to a power of n.

Abstract

We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^{\epsilon}} for some constant \epsilon>0, then with high probability the random graph G(n,p) contains a copy of T. The obtained bound on the edge probability is shown to be essentially tight for D=n^{\Theta(1)}.