Embedding induced trees in sparse expanding graphs

TL;DR

This paper introduces a new algorithm for embedding bounded degree induced trees in sparse expander graphs, extending previous results and establishing linear bounds for various graph properties with potential broad applications.

Contribution

The paper generalizes a key result to the induced setting and establishes linear bounds for induced Ramsey numbers of bounded degree trees.

Findings

Sparse random graphs contain all bounded degree trees of linear size whp

Induced and size induced Ramsey numbers of bounded degree trees are linear

Nearly-tight bounds on induced forests in bounded degree countable expanders

Abstract

Inspired by the network routing literature \cite{aggarwal1996efficient}, we develop what we call a ``Pre-Emptive Greedy Algorithm" to embed bounded degree induced trees in sparse expanders. This generalises a powerful and central result of Friedman and Pippenger to the induced setting. As corollaries we obtain that a sparse random graph contains all bounded degree trees of linear order (whp) and that the induced and size induced Ramsey numbers of bounded degree trees are linear. No such linear bounds were previously known. We also prove a nearly-tight result on induced forests in bounded degree countable expanders. We expect that our new result will find many more applications.