## Abstract

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k-1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices.

Original language | English (US) |
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Pages (from-to) | 389-410 |

Number of pages | 22 |

Journal | Combinatorica |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1992 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics