Embedding bounded degree spanning trees in random graphs

TL;DR

This paper proves that with high probability, a bounded degree spanning tree can be embedded in a specific random graph model, expanding understanding of graph embedding thresholds.

Contribution

It establishes a near-optimal threshold for embedding bounded degree spanning trees in random graphs, advancing theoretical knowledge in probabilistic combinatorics.

Findings

Bounded degree spanning trees can be embedded in (n,\u03b45 n/n) with high probability.

The result approaches the theoretical limit for such embeddings.

Provides new insights into the structure of random graphs and tree embeddings.

Abstract

We prove that if a tree $T$ has $n$ vertices and maximum degree at most $\Delta$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,\Delta\log^5 n/n)$.