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


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.

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Research supported by NSF grant CCR-8709818.

Research supported by NSF grant CCR-8805978 and Pennsylvania State University Research Initiation grant 428-45.

Research supported by NSF grants DCR-8603346 and CCR-8806308.

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Cai, JY., Fürer, M. & Immerman, N. An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389–410 (1992).

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