The specker-blatter theorem revisited

Authors:
E. Fischer;J. A. Makowsky
Affiliations:
Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa, IsraelHaifa, Israel;Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa, IsraelHaifa, Israel
Venue:
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Year:
2003

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Abstract

In this paper we study the generating function of classes of graphs and hypergraphs modulo a fixed natural number m. For a class of labeled graphs C we denote by fC(n) the number of structures of size n. For C definable in Monadic Second Order Logic MSOL with unary and binary relation symbols only, E. Specker and C. Blatter showed in 1981 that for every m ∈ N, fC(n) satisfies a linear recurrence relation fC(n) = Σj=1dm aj(m) fc (n - j), over Zm, and hence is ultimately periodic for each m. In this paper we show how the Specker-Blatter Theorem depends on the choice of constants and relations allowed in the definition of C. Among the main results we have the following: - For n-ary relations of degree at most d, where each element a is related to at most d other elements by any of the relations, a linear recurrence relation holds, irrespective of the arity of the relations involved. - In all the results MSOL can be replaced by CMSOL, Monadic Second Order Logic with (modular) Counting. This covers many new cases, for which such a recurrence relation was not known before.