## An Optimal Lower Bound on the Number of Variables for Graph Identification (1992)

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Venue: | Combinatorica |

Citations: | 135 - 9 self |

### BibTeX

@ARTICLE{Cai92anoptimal,

author = {Jin-yi Cai and Neil Immerman and Martin Fürer},

title = {An Optimal Lower Bound on the Number of Variables for Graph Identification},

journal = {Combinatorica},

year = {1992},

volume = {12},

pages = {389--410}

}

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### 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. 1