Why is the chromatic polynomial a polynomial?

Janos Makowsky

Sunday, 28.10.2007, 10:30

Room 337-8 Taub Bld.

The number of k-vertex colorings of a graph is a polynomial in k, the chromatic polynomial. Many other graph invariants (Tutte polynomial, matching polynomial, interlace polynomial, cover polynomial) are also polynomials (possibly in several variables). We show that this is no accident.
The speaker introduced the notion of graph polynomials definable in Monadic Second Order Logic, and showed that the Tutte polynomial and its generalization, the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. This definition can be extended to allow definability in full second order, or even higher order Logic.
In the analysis of the structure of totally categorical first order theories, B. Zilber (1981) showed that certain combinatorial counting functions play an important role. Those functions are invariants of the structures and are always polynomials in one or many variables, depending on the rank, respectively the independent dimension, of the theory in question.
The purpose of this talk is to show that many graph polynomials and combinatorial counting functions of graph theory do occur as combinatorial counting functions of totally categorical theory. We also give a characterization of polynomials definable in Second Order Logic.
The talk presents the results in an elementary way, although the underlying theory stems from deep results in model theory.
(Joint work with B. Zilber)