TY - JOUR

T1 - Large nearly regular induced subgraphs

AU - Alon, Noga

AU - Krivelevich, Michael

AU - Sudakov, Benny

PY - 2008

Y1 - 2008

N2 - For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n, 1) vertices. The problem of estimating f(n, 1) was posed long ago by Erdös, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of f(n, c): (i) For fixed c > 2.1, n1-0(1/c) ≤ f(n, c) ≤ O(cn/ logra). (ii) For fixed c = 1 + η with η > 0 sufficiently small, f(n,c) ≥ nΩ(η2 /ln(1/η)). (iii) Ω(lnra) ≤ f(n, 1) ≤ O(n1/2 ln3/4ra). An analogous problem for not necessarily induced subgraphs is briefly considered as well.

AB - For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n, 1) vertices. The problem of estimating f(n, 1) was posed long ago by Erdös, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of f(n, c): (i) For fixed c > 2.1, n1-0(1/c) ≤ f(n, c) ≤ O(cn/ logra). (ii) For fixed c = 1 + η with η > 0 sufficiently small, f(n,c) ≥ nΩ(η2 /ln(1/η)). (iii) Ω(lnra) ≤ f(n, 1) ≤ O(n1/2 ln3/4ra). An analogous problem for not necessarily induced subgraphs is briefly considered as well.

KW - Induced subgraphs

KW - Ramsey-type problem

KW - Regular subgraphs

UR - http://www.scopus.com/inward/record.url?scp=77950828318&partnerID=8YFLogxK

U2 - 10.1137/070704927

DO - 10.1137/070704927

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AN - SCOPUS:77950828318

SN - 0895-4801

VL - 22

SP - 1325

EP - 1337

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 4

ER -