Induced subgraphs with many distinct degrees

TL;DR

This paper proves that graphs with bounded clique or independent set size contain large induced subgraphs with many distinct degrees, extending previous results and establishing tight bounds for graphs with no small degree-distinct subgraphs.

Contribution

It generalizes earlier work by showing that graphs with small homogeneity contain large induced subgraphs with many degrees and provides tight bounds for graphs lacking small degree-distinct subgraphs.

Findings

Graphs with (hom(G)) contain induced subgraphs with ((n/ exthom(G))^{1/2}) distinct degrees.

For fixed k, graphs with no induced subgraph of k degrees have homogeneity at least n/(k-1)-o(n).

Bounds are essentially tight for the absence of small degree-distinct subgraphs.

Abstract

Let $\hom(G)$ 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 $\hom(G) = O(\log n)$ 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 $\hom(G) = O(n^\epsilon)$, 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 $\Omega((n/\hom(G))^{1/2})$ 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 $\hom(G) \ge n/(k-1)-o(n)$; this bound is essentially best-possible.