Embedding induced trees in sparse expanding graphs

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 corol...

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Bibliographic Details
Main Authors Girão, António, Hurley, Eoin
Format Journal Article
LanguageEnglish
Published 06.06.2024
Subjects
Mathematics - Combinatorics
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Summary: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.
DOI:10.48550/arxiv.2406.04260