Large nearly regular induced subgraphs

TL;DR

This paper investigates the size of large induced subgraphs with bounded degree ratios in graphs, providing bounds for different degree ratio parameters and extending to non-induced subgraphs.

Contribution

It establishes new asymptotic bounds for the maximum size of induced subgraphs with degree ratios, advancing understanding of regular and nearly regular subgraph existence.

Findings

For c>2.1, f(n,c) is between n^{1-O(1/c)} and O(cn/ log n).

For c=1+ε, f(n,c) grows at least as n^{Ω(ε^2/ log(1/ε))}.

For c=1, f(n,1) is between Ω(log n) and O(n^{1/2} log^{3/4} n).

Abstract

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so 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 $(n,1) was posed long time ago by Erdos, Fajtlowicz and Staton. In this note 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+\epsilon with epsilon>0 sufficiently small, f(n,c) \geq n^{\Omega(\epsilon^2/ \ln (1/\epsilon))}. (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.