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

    253 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 1 1992

    All Science Journal Classification (ASJC) codes

    • Discrete Mathematics and Combinatorics
    • Computational Mathematics

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