## Abstract

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

Original language | English (US) |
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Pages (from-to) | 256-271 |

Number of pages | 16 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 163 |

DOIs | |

State | Published - Nov 2023 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Induced subgraphs
- Sparse graphs