Strengthening Rodl's theorem

TL;DR

This paper enhances Rodl's theorem by proving that any H-free graph can be partitioned into a fixed number of parts, each inducing a c-restricted graph, thus providing a more comprehensive structural understanding.

Contribution

It extends Rodl's original result by showing a bounded partition into c-restricted subgraphs for H-free graphs, regardless of the size.

Findings

Every H-free graph can be partitioned into a bounded number of c-restricted subgraphs.

The result applies for all graphs H and all positive c values.

Provides a stronger structural decomposition than previous theorems.

Abstract

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs.