Induced subgraphs with many distinct degrees

Abstract

Let $\mathrm{hom}(G)$ denote the size of the largest clique or independent set of a graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$ with $\mathrm{hom}(G)=O(\log n)$ contains an induced subgraph with $\Omega (n^{1/2})$ distinct degrees, and raised the question of deciding whether an analogous result holds for every $n$-vertex graph $G$ with $\mathrm{hom}(G)=O(n^{\epsilon })$, where $\epsilon >0$ is a fixed constant. Here, we answer their question in the affirmative and show that every graph $G$ on $n$ vertices contains an induced subgraph with $\Omega ((n/\mathrm{hom}(G){)}^{1/2})$ distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed $k\in \mathbb{N}$, that if an $n$-vertex graph $G$ contains no induced subgraph with $k$ distinct degrees, then $\mathrm{hom}(G)\ge n/(k-1)-o(n)$; this bound is essentially best-possible.

17 pages, Combinatorics, Probability and Computing