On Counting Generalized Colorings

The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorings. We show that this class encompasses the examples of graph polynomials from the literature. Furthermore, we extend the definition of graph polynomials definable in MSOL to allow definability in full second order, SOL. Finally, we show that the SOL-definable graph polynomials extended with a combinatorial counting function are exactly the counting functions of generalized colorings definable in SOL. ? Partially supported by the Israel Science Foundation for the project ”Model Theoretic Interpretations of Counting Functions” (2007-2010) and the Grant for Promotion of Research by the Technion–Israel Institute of Technology. ?? Partially supported by MODNET: Marie Curie Research Training Network MRTNCT-2004-512234