## Abstract

The regularity lemma of Szemerédi gives a concise approximate description of a graph via a so-called regular partition of its vertex set. In this paper we address the following problem: Can a graph have two "distinct" regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand, we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very "similar." En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rödl and Schacht and by Lovász and Szegedy from a previously known variant of the regularity lemma due to Alon et al. in 2000. The proof also provides a deterministic polynomial time algorithm for finding such partitions.

Original language | English (US) |
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Pages (from-to) | 278-287 |

Number of pages | 10 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- Algorithm
- Isomorphic
- Regularity lemma