An optimal lower bound on the number of variables for graph identification

Jin Yi Cai, Martin Fürer, Neil Immerman

Research output: Contribution to journalArticlepeer-review

396 Scopus citations

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 languageEnglish (US)
Pages (from-to)389-410
Number of pages22
JournalCombinatorica
Volume12
Issue number4
DOIs
StatePublished - Dec 1992

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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