A Tutte polynomial for maps II: The non-orientable case

Abstract

We construct a new polynomial invariant of maps (graphs embedded in a closed surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás—Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.

Introduction

In [41] Tutte defined the dichromate of a graph $\Gamma$ as a bivariate polynomial graph invariant that includes the chromatic polynomial of $\Gamma$ and the flow polynomial of $\Gamma$ as univariate specializations. The latter two polynomials can be (and usually are) defined by their evaluations at positive integers. Let ${\mathbb{Z}}_n$ denote the additive cyclic group of order $n$ and suppose we fix an arbitrary orientation of the edges of $\Gamma$. A nowhere-zero ${\mathbb{Z}}_n$-flow of $\Gamma$ is an assignment of non-zero elements of ${\mathbb{Z}}_n$ to the edges of $\Gamma$ such that Kirchhoff’s law is satisfied at each vertex; that is the sum of the values on the incoming edges equals the sum of the values of the outgoing edges. (It then follows that for any edge cutset the sum of the values on edges in one direction is equal to the sum of values on edges in the other direction. It is also evident that the number of nowhere-zero ${\mathbb{Z}}_n$-flows is an invariant of the graph $\Gamma$, as this number does not depend on the choice of orientation of edges of $\Gamma$.) For each positive integer $n$, the flow polynomial of $\Gamma$ evaluated at $n$ is equal to the number of nowhere-zero ${\mathbb{Z}}_n$-flows of $\Gamma$. The chromatic polynomial evaluated at $n$ counts the number of proper $n$-colorings of $\Gamma$. A proper coloring of $\Gamma$ induces a nowhere-zero ${\mathbb{Z}}_n$-tension of $\Gamma$, which is to say an assignment of non-zero elements of ${\mathbb{Z}}_n$ to edges of $\Gamma$ such that, for each closed walk, the sum of the values on forward edges equals the sum of the values on backward edges. Upon fixing the color of a vertex in each connected component of $\Gamma$ there is a one-to-one correspondence between proper $n$-colorings and nowhere-zero ${\mathbb{Z}}_n$-tensions of $\Gamma$.

The dichromate was to become better known as the Tutte polynomial and not only contains as evaluations many other important graph invariants, but also extends its domain from graphs to matroids, and has revealed fruitful connections between graphs and many other combinatorial structures, such as the Potts model of statistical physics and, more topologically, knot diagrams. Another natural way to extend the domain of graphs is to maps, that is, graphs embedded in a closed surface (an orientable map if the surface is orientable, and a non-orientable map otherwise). Local flows and local tensions of a map are defined similarly to flows and tensions of a graph, and coincide for a plane map with flows and tensions of the underlying planar graph. Furthermore, values in a local flow may be taken from a nonabelian group, as the cyclic ordering of edges around vertices of a map determines in which order to multiply elements together when verifying that Kirchhoff’s law holds. (We adopt the convention that in nonabelian groups composition is multiplication, while in abelian groups composition is addition and the identity is zero.) Just as the flow polynomial of a graph evaluated at a positive integer $n$ is equal to the number of nowhere-zero ${\mathbb{Z}}_n$-flows, so for each finite group $G$ we have a map invariant equal to the number of nowhere-identity local $G$-flows. Local tensions are defined dually (just facial walks rather than all closed walks being involved in the definition: the correspondence of tensions with vertex colorings is not preserved, but see [33], [34]). The question is then whether there is a polynomial map invariant which contains as evaluations the number of nowhere-identity local $G$-flows and the number of nowhere-identity local $G$-tensions, in a similar way to how the Tutte polynomial of a graph contains the flow polynomial and the chromatic polynomial as specializations.

Various extensions of the Tutte polynomial to maps have been defined, notably by Las Vergnas [31], Bollobás and Riordan [1], [2], and Krushkal [5], [30], each of which have properties analogous to those of the Tutte polynomial such as having a deletion–contraction recurrence formula or extending from graphs to matroids (the relevant extension from maps being to $\Delta$-matroids [6]). However, none of these extensions of the Tutte polynomial is known to contain for every finite group $G$ the number of nowhere-identity local $G$-flows and the number of nowhere-identity local $G$-tensions as evaluations. Recently such an extension of the Tutte polynomial to orientable maps, called the surface Tutte polynomial, was discovered by three of the authors together with Krajewski [17]. The surface Tutte polynomial of an orientable map includes the Krushkal polynomial of an orientable map, and hence the Las Vergnas polynomial and Bollobás–Riordan polynomial of an orientable map, as specializations. We remark here that throughout this paper a map is a graph which is 2-cell embedded in a closed surface (either orientable or non-orientable); if the graph is not connected, then each connected component is embedded in its own closed surface.

In the present paper we extend the domain of the surface Tutte polynomial of [17] to include non-orientable maps and show that this map invariant contains for every finite group $G$ the number of nowhere-identity local $G$-flows and the number of nowhere-identity local $G$-tensions as evaluations, as well as containing further specializations such as the number of spanning quasi-trees of given genus. The surface Tutte polynomial of a map can therefore be seen as the analogue of the dichromate of a graph, which was defined by Tutte to include the number of nowhere-zero ${\mathbb{Z}}_n$-flows and the number of nowhere-zero ${\mathbb{Z}}_n$-tensions as evaluations. Furthermore, the Kruskhal polynomial of a non-orientable map (as defined by Butler [5]) remains a specialization of the surface Tutte polynomial extended to maps. An extended abstract of this paper appeared in the proceedings of EUROCOMB 2017 [18].

While extending the surface Tutte polynomial of [17] from orientable maps to non-orientable maps suggested itself as a natural step to take, and the theorems we prove are generalizations of theorems in [17], in each case the added complications of non-orientability necessitated a substantial development of technique in order to achieve the required lifting of a theorem about orientable maps to a theorem about maps. (A like remark applies to Bollobás and Riordan’s extension of their polynomial from orientable maps [1] to include non-orientable maps [2].)

While the surface Tutte polynomial is not itself an invariant of the underlying $\Delta$-matroid of a map, it contains specializations that do have this property, including, apart from the Krushkal polynomial, an as yet unstudied four-variable $\Delta$-matroid invariant that we introduce in Section 5.2. (In this respect the surface Tutte polynomial is similar to the $U$-polynomial of Noble and Welsh [38], a graph invariant in infinitely many variables, which is not itself an invariant of the underlying matroid of the graph, even though it contains many such matroid invariants as specializations, including the Tutte polynomial.)

In Section 2 we start by viewing maps in the conventional way as graphs embedded in closed surfaces and introduce numerical map parameters such as the genus. We then proceed to describe Tutte’s permutation axiomatization of maps, which may be less familiar to the reader but proves well suited to the study of local flows and local tensions of maps. To keep this section as brief as possible, the definition of deletion and contraction of edges in maps using this formalism is deferred to Appendix A.

The subject of this paper, the surface Tutte polynomial for maps, is introduced in Section 3. We derive some of its elementary properties and show that it includes Butler’s extension [5] of the Krushkal polynomial [30] to maps (non-orientable as well as orientable).

In Section 4 we use Tutte’s permutation axiomatization of maps to give a streamlined definition of local flows and local tensions of maps taking values in a finite group. A key result of this paper is Theorem 4.6, giving an explicit formula for the number of local flows of a map taking non-identity values in a given finite group. The main steps in the proof of this theorem are given in Section 4.4, after stating some of its immediate corollaries in Section 4.3, including those evaluations of the surface Tutte polynomial that give the number of nowhere-identity local flows and number of nowhere-identity local tensions of a map. One of the two key ingredients needed in the proof of Theorem 4.6 is a combinatorial version of the classification theorem for closed surfaces (stated as Theorem 4.17). We used the recent book [15] as a reference for this classical theorem and for the language of cell complexes that is needed to make use of it. In Appendix C we translate the language of cell complexes to Tutte’s axiomatization of maps. The other key ingredient is a result on counting homomorphisms from the fundamental group of a surface to a given finite group (stated as Theorem 4.14). This result can be found in the literature but may not be readily accessible to combinatorialists. We have therefore included a proof in Appendix B that only uses elementary representation theory.

Besides the number of nowhere-identity local flows and nowhere-identity local tensions the surface Tutte polynomial contains other significant map invariants as specializations. In Section 5 we consider evaluations analogous to those of the Tutte polynomial of a connected graph that enumerate spanning trees, spanning forests and connected spanning subgraphs. In this section we also introduce a different normalization of the surface Tutte polynomial in Proposition 5.1 and a four-variable specialization of it in Definition 5.6 similar in form to the Krushkal polynomial.