Large Nearly Regular Induced Subgraphs

Abstract

For a real $c \geq 1$ 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 Erdos, 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 $c>2.1$, $n^{1-O(1/c)} \leq f(n,c) \leq O(cn/\log n)$. (ii) For fixed $c=1+\varepsilon$ with $\varepsilon>0$ sufficiently small, $f(n,c) \geq n^{\Omega(\varepsilon^2/ \ln (1/\varepsilon))}$. (iii) $\Omega (\ln n) \leq f(n,1) \leq O(n^{1/2} \ln^{3/4} n)$. An analogous problem for not necessarily induced subgraphs is briefly considered as well.