Large Nearly Regular Induced Subgraphs

For a real c ≥ 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 f(n, 1) was posed long time ago by Erdős, 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 ≤ f(n, c) ≤ O(cn/ logn). (ii) For fixed c = 1 + ε with ε > 0 sufficiently small, f(n, c) ≥ nΩ(ε2/ . (iii) Ω(lnn) ≤ f(n, 1) ≤ O(n ln n). An analogous problem for not necessarily induced subgraphs is briefly considered as well.