TY - JOUR

T1 - Finding and using expanders in locally sparse graphs

AU - Krivelevich, Michael

N1 - Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Expanders

KW - Graph minors

KW - Positional games

KW - Random graphs

UR - http://www.scopus.com/inward/record.url?scp=85045647414&partnerID=8YFLogxK

U2 - 10.1137/17M1128721

DO - 10.1137/17M1128721

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AN - SCOPUS:85045647414

SN - 0895-4801

VL - 32

SP - 611

EP - 623

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -