Graph colorings, flows and arithmetic Tutte polynomial

Abstract

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte (1954) [9].It is well known how to associate a matroid, and hence a Tutte polynomial, to a graph. Moreover, several combinatorial objects associated to a graph are counted by suitable specializations of this important invariant: (proper) q-colorings and (nowhere zero) q-flows are two of the most classical examples. See [6] and [10] for systematic accounts, or [1] for more recent developments.In [7] a new polynomial has been introduced, which provides a natural counterpart for toric arrangements of the Tutte polynomial of a hyperplane arrangement. In fact in [7] and [3] it has been shown how this polynomial has several applications to toric arrangements, vector partition functions and zonotopes.In [4] we introduced the notion of an arithmetic matroid, which generalizes that of a matroid, and whose main example is provided by a list of elements in a finitely generated abelian group. To this object we associated an arithmetic Tutte polynomial (which is the polynomial in [7] for the main example) and provided a combinatorial interpretation of it which extends the one given by Crapo for the classical Tutte polynomial.Encouraged by all this evidence (cf. also [5] where two parallel theories are developed), we think of the arithmetic matroids and the arithmetic Tutte polynomial as natural generalizations of their classical counterparts. So it seemed also natural to us to look for applications in graph theory.

The multivariate arithmetic Tutte polynomial

“…Let us see how colorings and flows on graphs may be generalized using the ideas in the previous sections. Some of the following observations were also made in [8], where the results were obtained by deletion-contraction arguments.…”

“…We have the following result, generalizing a famous theorem of Tutte [13], and also its analogue in the arithmetic setting [8]: Proof. The first statement follows immediately from (7.3), since χ L (q) = Z P L (q, −1).…”

“…This is a family of submanifolds in the abelian compact Lie group Hom(G, S 1 ) (see [10,Section 5]); • the dimension of the Dahmen-Micchelli space DM (L). This vector space was introduced in order to study vector partition functions (see [10,Section 6]); • the Ehrhart polynomial of the zonotope Z(L) (see [10,Section 4] and [8]). Furthermore, the arithmetic Tutte polynomial has applications to graph theory [8], and it has nonnegative coefficients, as proved in [7] by providing a combinatorial interpretation that extends Crapo's theorem.…”

Universal Tutte characters via combinatorial coalgebras

“…We only mention one that was proved by Backman and Lenz [3]. This follows from formula (21) in the case (m 1 , m 2 ) = (m, 1) or (1, m) by the projection Z >0 → 1.…”

Matroids over a ring

“…It thus comes as no surprise that the Tutte polynomial has been considered for generalizations of matroids as well. A quasi-arithmetic matroidM has an associated arithmetic Tutte polynomial MM (x, y), which has proved to be an useful tool in studying toric arrangements, partition functions, zonotopes, and graphs ( [17,7,3]). More strongly, the authors of [3] define a Tutte quasi-polynomial of an integer vector configuration, interpolating between T M (x, y) and MM (x, y), which is no longer an invariant of the quasi-arithmetic matroid (as it depends on the groups, not just their cardinalities).…”