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 -