Finding and using expanders in locally sparse graphs

Michael Krivelevich*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants c1 > c2 > 1, 0 < α < 1, a graph G on n vertices is called a (c1, c2, α)-graph if it has at least c1n edges, but every vertex subset W ⊂ V (G) of size |W| ≤ αn spans less than c2|W| edges. We prove that every (c1, c2, α)-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.

Original languageEnglish
Pages (from-to)611-623
Number of pages13
JournalSIAM Journal on Discrete Mathematics
Issue number1
StatePublished - 2018


FundersFunder number
USA-Israel BSF2014361
Israel Science Foundation1261/17


    • Expanders
    • Graph minors
    • Positional games
    • Random graphs


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