, Volume 12, Issue 4, pp 389–410 | Cite as

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

  • Jin-Yi Cai
  • Martin Fürer
  • Neil Immerman


In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-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 becausen variables obviously suffice to identify graphs onn vertices.

AMS subject classification code (1991)

03 B 10 05 C 60 05 C 85 


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Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Martin Fürer
    • 2
  • Neil Immerman
    • 3
  1. 1.Computer Science Dept.Princeton UniversityPrinceton
  2. 2.Computer Science Dept.University of MassachusettsAmherst
  3. 3.Computer Science Dept.Pennsylvania State UniversityUniversity Park

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