Induced subgraphs of prescribed size

Noga Alon; Michael Krivelevich; Benny Sudakov

doi:10.1002/jgt.10117

Induced subgraphs of prescribed size

Noga Alon, Michael Krivelevich, Benny Sudakov

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdos and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n^δ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e^{Ω(√log n)}.

Original language	English
Pages (from-to)	239-251
Number of pages	13
Journal	Journal of Graph Theory
Volume	43
Issue number	4
DOIs	https://doi.org/10.1002/jgt.10117
State	Published - 1 Jan 2003
Externally published	Yes

Keywords

Cliques and independent sets
Induced subgraphs
Ramsey graphs

ASJC Scopus subject areas

Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1002/jgt.10117

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abstract = "A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdos and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and nδ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and eΩ(√log n).",

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AU - Alon, Noga

AU - Krivelevich, Michael

AU - Sudakov, Benny

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N2 - A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdos and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and nδ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and eΩ(√log n).

AB - A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdos and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and nδ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and eΩ(√log n).

KW - Cliques and independent sets

KW - Induced subgraphs

KW - Ramsey graphs

UR - https://www.scopus.com/pages/publications/0043240128

U2 - 10.1002/jgt.10117

DO - 10.1002/jgt.10117

M3 - Article

AN - SCOPUS:0043240128

SN - 0364-9024

VL - 43

SP - 239

EP - 251

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 4

ER -

Induced subgraphs of prescribed size

Abstract

Keywords

ASJC Scopus subject areas

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