Sharp threshold for embedding combs and other spanning trees in random graphs

TL;DR

This paper establishes a sharp threshold for embedding combs and certain spanning trees in random graphs, and introduces an efficient method for finding large expander subgraphs, improving existing results.

Contribution

It proves a sharp threshold for embedding combs and related trees in random graphs, and provides an efficient method for locating large expander subgraphs.

Findings

Almost surely contains combs at (1+ε)log n/n threshold

Improves thresholds for embedding spanning trees with disjoint paths

Provides an efficient method for finding large expander subgraphs

Abstract

When $k|n$, the tree $\mathrm{Comb}_{n,k}$ consists of a path containing $n/k$ vertices, each of whose vertices has a disjoint path length $k-1$ beginning at it. We show that, for any $k=k(n)$ and $\epsilon>0$, the binomial random graph $\mathcal{G}(n,(1+\epsilon)\log n/ n)$ almost surely contains $\mathrm{Comb}_{n,k}$ as a subgraph. This improves a recent result of Kahn, Lubetzky and Wormald. We prove a similar statement for a more general class of trees containing both these combs and all bounded degree spanning trees which have at least $\epsilon n/ \log^9n$ disjoint bare paths length $\lceil\log^9 n\rceil$. We also give an efficient method for finding large expander subgraphs in a binomial random graph. This allows us to improve a result on almost spanning trees by Balogh, Csaba, Pei and Samotij.