Tutte's dichromate for signed graphs
Andrew Goodall, Bart Litjens, Guus Regts, Lluis Vena
TL;DR
This paper introduces the trivariate Tutte polynomial for signed graphs, unifying various graph invariants and extending to matroid pairs, revealing new relationships and differences with existing polynomials.
Contribution
It defines a new invariant, the trivariate Tutte polynomial for signed graphs, and explores its properties, extensions, and distinctions from related polynomials.
Findings
The trivariate Tutte polynomial encodes proper colorings and flows of signed graphs.
It differs from the dichromatic polynomial in capturing nowhere-zero flows.
The polynomial extends to matroid pairs, linking graph theory and matroid theory.
Abstract
We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial)…
| ordinary | edge not a loop or bridge in that when removed does not change the number of balanced connected components | ordinary | ordinary |
| ordinary | edge not a loop in that belongs to every unbalanced cycle in its connected component (of which there is at least one) | ordinary | coloop |
| bridge | bridge of in an unbalanced loose handcuff | coloop | ordinary |
| bridge | bridge of in no unbalanced loose handcuff | coloop | coloop |
| loop | loop of with negative sign, other unbalanced cycles in its connected component | loop | ordinary |
| loop | loop of with negative sign, no other unbalanced cycles in its connected component | loop | coloop |
| loop | loop of with positive sign | loop | loop |
| eq1 | eq2 | other cycles | |||
|
( sym.) |
NA | ||||
| NA | |||||
| NA | |||||
|
( sym.) |
|||||
| eq1 | eq2 | other cycles | |||
|
( sym.) |
( sym.) |
||||
| NA | |||||
| eq1 | eq2 | other cycles | |||
| NA | |||||
| NA | |||||
| NA | |||||
| NA | |||||
| NA | |||||
| eq1 | eq2 | other cycles | |||
|---|---|---|---|---|---|
| NA | |||||
| NA | |||||
| NA | |||||
| eq1 | eq2 | other cycles | |||
| Name | Expression | Reference | ||
|---|---|---|---|---|
|
Tutte poly. graph
|
(2) | |||
|
Tutte poly.
matroid |
(4) | |||
|
trivar. Tutte poly.
pair of matroids |
(46) | |||
|
linking poly.
pair of matroids |
|
|||
|
Las Vergnas poly.
mat. persp. |
[31, (5.2)] | |||
|
trivar. Tutte
signed graph |
(5) | |||
|
flow polynomial
signed graph finite abelian group |
(24) | |||
|
chromatic poly.
signed graph |
(26) | |||
|
nz pot. diff.
signed graph finite abelian group |
(41) | |||
|
surface Tutte poly.
map |
[20, (4)] | |||
|
Krieger–O’Connor poly.
signed graph |
[27] |
| Name | Name | Relation | Ref. | ||
|
trivar. Tutte poly.
pair of matroids |
Las Vergnas poly.
matroid persp. |
(54) | |||
|
trivar. Tutte poly.
signed graph |
trivar. Tutte poly.
pair of matroids |
|
|||
|
linking poly.
pair of matroids |
trivar. Tutte
pair of matroids |
|
(49) | ||
|
flow polynomial
signed graph fin. ab. group |
trivar. Tutte poly.
signed graph |
|
(25) | ||
|
trivar. Tutte poly.
signed graph |
surface Tutte poly.
map |
|
(8) | ||
|
Krieger–O’Connor poly.
signed graph |
trivar. Tutte poly.
signed graph |
Tab 8 |
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Tutte’s dichromate for signed graphs
Andrew Goodall Corresponding author. Charles University, Prague, Czech Republic. Email: [email protected]. Supported by Project ERCCZ LL1201 Cores and Czech Science Foundation GA ČR 19-21082S.
Bart Litjens University of Amsterdam, Netherlands. Email: [email protected]. Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n 339109.
Guus Regts University of Amsterdam, Netherlands. Email: [email protected]. Supported by a NWO Veni grant.
Lluís Vena University of Amsterdam, Netherlands. Email: [email protected]. Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n 339109.
Abstract
We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set – for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.
Keywords
signed graph, Tutte polynomial, flow, tension, coloring, matroid
1 Introduction
Signed graphs, introduced by Harary [21], are graphs (loops and multiple edges allowed) in which each edge is given a positive or negative sign. A large literature has accumulated on signed graphs [45]. Notably, Zaslavsky [40, 41, 42, 43] developed the theory of signed graphs with respect to colorings, orientations and matroids associated with signed graphs. Just as colorings, flows and orientations of a graph may be defined with reference solely to the underlying cycle matroid of the graph, so may colorings, flows and orientations of a signed graph be defined in terms of the underlying frame matroid of the signed graph.
For a finite additive abelian group , a (nowhere-zero) -flow of a graph is defined with reference to an arbitrary orientation of the graph as an assignment of (non-zero) elements of to its edges such that Kirchhoff’s law is satisfied, i.e. for each vertex the sum of values on its incoming edges is equal to the sum of values on its outgoing edges. The Tutte polynomial of a graph includes as specializations the chromatic polynomial, counting proper vertex colorings, and the flow polynomial, counting nowhere-zero -flows (the number of which is independent of the orientation of the graph needed to define a flow, and, as Tutte showed [38], depends only on ). To a (proper) vertex coloring of a graph using at most colors corresponds, by taking the “potential difference” at the ends of an edge, a (nowhere-zero) -tension of the graph, where is an additive abelian group or order ; conversely, for a connected graph there are exactly (proper) vertex colorings for any given (nowhere-zero) -tension. (See the paragraph on tensions and colorings below for how -tensions are defined.)
Tutte [38] showed how the dichromatic polynomial of a graph (Whitney rank generating function) is equivalent to the dichromate of the graph (specializing to the chromatic polynomial and, dually, to the flow polynomial). Zaslavsky [44] defines the dichromatic polynomial of a biased graph (of which a signed graph is a special case) and develops analogous properties to the dichromatic of a graph. In contrast to the case of graphs, however, Zaslavsky’s bivariate dichromatic polynomial of a signed graph does not qualify as being the dichromate of a signed graph since it does not in general yield the number of nowhere-zero flows as an evaluation. In order to enumerate both flows and colorings of signed graphs the dichromatic is not sufficient: the trivariate Tutte polynomial of this paper is needed, which contains the dichromatic polynomial as a specialization (see Section 3.2 below).
Our main results are that, similarly to the Tutte polynomial of a graph, the trivariate Tutte polynomial of a signed graph includes among its evaluations the number of (nowhere-zero) -flows and the number of proper -colorings for a given finite additive abelian group . (The points of evaluation depend on and , where .) Furthermore, we show that the number of nowhere-zero -tensions of a signed graph can be obtained from its trivariate Tutte polynomial evaluated at two different points (dependent on and ). For signed graphs, the number of (nowhere-zero) -tensions is not simply related to the number of (proper) -colorings: there is a group homomorphism from -colorings to -tensions, as for graphs, but in the case of signed graphs this homomorphism is not in general surjective except when is odd.
Flows
Flows are defined for signed graphs in a similar way to graphs [6] via Kirchhoff’s law. The number of nowhere-zero -flows of a signed graph, however, depends on the number of elements of order in (the order of its 2-torsion subgroup). For a finite additive abelian group with 2-torsion subgroup of order , Beck and Zaslavsky [1] showed that when (so is odd), the number of nowhere-zero -flows of a given signed graph is a polynomial in , given as an evaluation of the Tutte polynomial of the underlying frame matroid. DeVos et al. [13], by establishing a deletion-contraction recurrence for the number of nowhere-zero -flows reducing its evaluation to single-vertex signed graphs consisting solely of loops, showed that it is a polynomial in and . In particular, this number is not an evaluation of the Tutte polynomial of the underlying frame matroid (unless is odd or the signed graph is balanced). In this paper we show that the number of nowhere-zero -flows is an evaluation of the trivariate Tutte polynomial, thereby establishing a subset expansion for this number (Corollary 5.2). While this paper was in preparation, this expansion was found independently and using different methods by Qian and Ren [36, Corollary 4].
Tensions and colorings
In Section 6 we begin by reviewing Zaslavsky’s enumeration of proper -colorings and proper non-zero -colorings of a signed graph, and show that these are evaluations of our signed graph Tutte polynomial: the number of proper -colorings is an evaluation of the Tutte polynomial of the underlying frame matroid, while the number of proper non-zero -colorings is not such an evaluation. Zaslavsky’s definition of signed graph proper -colorings, in which colors are elements of , has a natural generalization to proper -colorings, in which colors are taken from a finite set equipped with an involution : we enumerate these colorings, showing that they are evaluations of the trivariate Tutte polynomial at points dependent on and the number of elements of that are fixed by (Theorem 6.6). When is the set of elements of a finite additive abelian group , we take the involution to be negation: fixed points of the involution are elements of of order 1 or 2, and a proper -coloring is one that for each edge assigns colors and to its endpoints so that () when the edge is positive (negative).
For graphs, -tensions are assignments of elements of an additive abelian group to edges with the property that the cumulative sum of edge values encountered when traversing a closed walk is zero, where edge values are negated when the direction of the walk opposes the orientation of the edge. This definition may be more compactly phrased in terms of oriented circuits of the underlying graphic matroid. For signed graphs, a similar definition applies, with an orientation of an edge consisting in orienting its two half-edges, and only closed walks traversing an even number of negative edges being considered (see Figure 4). Likewise, the definition of a signed graph tension may be more compactly formulated in terms of oriented circuits of the underlying frame matroid, as is done in [9].
The relation of signed graph tensions to signed graph vertex colorings (see [40, 41]) is not as straightforward as that of graph tensions to graph vertex colorings. In a similar way to graphs, each (proper) -coloring of a signed graph yields a (nowhere-zero) -tension, in which the value on an edge is given by taking the difference (sum) of the colors of its endpoints when the edge is positive (negative). In general, however, not every nowhere-zero -tension arises from a -coloring in this way: tensions that do arise from -colorings will be called -potential differences. (For graphs, potential differences and tensions coincide.) In Section 7 we give an equivalent definition of -potential differences that refers only to the underlying frame matroid of the signed graph, i.e. independent of coloring vertices. Given the correspondence between (nowhere-zero) -potential differences and (proper) -colorings, we can use the previous enumeration of (proper) -colorings to deduce directly that the number of (nowhere-zero) -potential differences is given by an evaluation of the trivariate Tutte polynomial. In Section 7 we also establish that while the number of -tensions (edges allowed to take value [math]) is an evaluation of the trivariate Tutte polynomial, the number of nowhere-zero -tensions is generally not given by an evaluation of the trivariate Tutte polynomial at a single point. For a connected signed graph, the number of nowhere-zero -tensions is, however, up to a sign depending on the nullity of the underlying graph, equal to a linear combination of evaluations of the trivariate Tutte polynomial at two different points (dependent on and ), where the prefactors of these two evaluations depend only on and on whether the signed graph is balanced or unbalanced.
Matroids
Much as the Tutte polynomial of a graph depends only on its underlying cycle matroid, the trivariate Tutte polynomial of a signed graph depends only on its frame matroid and the cycle matroid of its underlying graph. The number of nowhere-zero flows and nowhere-zero tensions of a signed graph in general depend both on its underlying cycle matroid and on its frame matroid, and it is for this reason the Tutte polynomial of the frame matroid (equivalent to the dichromatic polynomial [44]) fails to give them. The trivariate Tutte polynomial of a signed graph extends to a polynomial invariant of arbitrary pairs of matroids on a common ground set, which includes the Las Vergnas polynomial [30] in the special case of matroid perspectives and is equivalent to the four-variable linking polynomial of a pair of matroids of Welsh and Kayibi [39], as explained in Appendix A.
Krajewski et al. [26], using a Hopf algebra framework, unify many of the existing extensions of the Tutte polynomial of a graph to other combinatorial objects (for example, those of Las Vergnas [30], Bollobás and Riordan [4, 5], Krushkal [28], and Butler [8] to embedded graphs, that of Crapo to matroids [12], and that of -matroids by Chun et al. [11]). Dupont et al. [14] extend this approach to bialgebras more generally. For a given graded class of combinatorial objects with a notion of deletion and contraction, there is a canonically defined invariant that shares with the Tutte polynomial of a graph the property of satisfying a deletion-contraction recurrence terminating in trivial objects, being universal for deletion-contraction invariants on the class, having a subset sum expansion, and satisfying a duality formula. This canonical invariant accordingly merits the designation of being the Tutte polynomial for the class. The trivariate Tutte polynomial defined in this paper is in this sense the canonically defined Tutte polynomial for the class of equivalence classes of signed graphs under switching (flipping signs of edges incident with a common vertex, a loop preserving its sign under this operation).
Summary of enumerative results
Table 1 below summarizes the main enumerative results of this paper given by evaluations of the trivariate Tutte polynomial, placed alongside the comparable evaluations of the Tutte polynomial for graphs. In it, is a graph and is a signed graph with signature , is a finite additive abelian group, and .
Organization
In the opening Section 2 we briefly introduce signed graphs and their underlying graphic and frame matroid, singling out those properties required in the sequel. In Section 3 we introduce the trivariate Tutte polynomial of a signed graph via its spanning subgraph expansion; in Appendix A we explain how this invariant of signed graphs is a special case of the trivariate Tutte polynomial of a pair of matroids on a common ground set, obtained by taking the matroids to be the underlying graphic matroid and frame matroid of the signed graph. We end Section 3 by connecting the trivariate Tutte polynomial of a signed graph to other related polynomials defined in the literature. (In Appendix C we give an overview of many of these polynomials.) In Section 4 we show how the trivariate Tutte polynomial behaves under deletion-contraction, and give a “Recipe Theorem” for the polynomial for signed graphs, which is the key result for proving the enumerations in Sections 5–7 given as evaluations of the trivariate Tutte polynomial (see Table 1 above).
2 Signed graphs and their matroids
2.1 Signed graphs
A signed graph is a pair , where is a finite undirected graph possibly with loops and multiple edges, called the underlying graph of , and is a function that associates a sign to each edge of , called the signature of . A cycle in is called balanced in if and unbalanced otherwise. The signed graph is itself called balanced if each cycle of is balanced in and unbalanced otherwise.
For a signed graph , we define , the number of connected components of the underlying graph . The number of balanced and unbalanced connected components of are denoted by and by , respectively. Thus .
Switching at a vertex means negating the sign of every edge that is incident with , while keeping the sign of each loop attached to . We say that two signed graphs and are equivalent if the graph is isomorphic to the graph , and if, under such an isomorphism, the signature can be obtained from by a sequence of switchings at vertices. The property of being balanced or unbalanced is constant on equivalence classes of signed graphs, as being balanced or unbalanced is invariant under switching at a vertex.
The deletion of an edge in yields the signed graph , where is the restriction of to and where is the graph obtained from by deleting as a graph edge. This signed graph obtained from be deleting is denoted by , and the signed graph obtained by deleting all the edges in is denoted by . For we write for the complementary subset ; the signed graph is restriction of to .
The contraction of a non-loop edge of that has positive sign in yields the signed graph , where is the restriction of to and where is the graph obtained from by contracting as a graph edge. The signed graph obtained from by contracting is denoted by . Note that by switching we can always ensure that the sign of a non-loop edge is positive. When is a loop with positive sign in we set . In order to define contraction of negative loops, Zaslavsky [41] enlarges the definition of signed graphs to include half-arcs and free loops. In our case, as we can avoid contracting negative edges and loops, we do not define the contraction by an arbitrary subset of edges , only for subsets of positive edges (after possible switching – Zaslavsky [42] shows that the order in which the edges are contracted does not affect the outcome).
2.2 Matroids
For further background on matroids see [35]. Here we highlight what is needed in the sequel.
A matroid may – among many “cryptomorphic” axiomatizations – be defined in terms of its collection of independent sets, by its collection of bases (independent sets of maximum size), by its circuits (minimal dependent sets), or by its rank function (size of a maximal independent subset).
A coloop of a matroid is defined by the property that it belongs to no circuit; equivalently, a coloop belongs to every basis. A loop of a matroid has the defining property that it belongs to no independent set of edges, and in particular to no basis. An element of a matroid that is neither a loop nor a coloop is ordinary.
2.2.1 The cycle matroid of a graph
To a graph there corresponds a matroid on ground set with rank function defined for by (equal to the size of a maximal spanning forest of ). For a connected graph , the bases of are the edge sets of spanning trees of , the independent sets are the edge sets of spanning forests of , and the dependent sets the edge sets of spanning subgraphs containing a cycle of . The circuits (minimal dependent sets) are edge sets of spanning subgraphs minimal with respect to containing a cycle in , called circles in [42].
A coloop in the matroid is a bridge of (deleting a bridge increases the number of connected components by one). A loop in is an edge of with .
2.2.2 The frame matroid of a signed graph
The material in this section is drawn from Chapter 6.10 of [35].
A subdivision of the left-hand graph in Figure 1 is a tight handcuff, and a subdivision of the right-hand graph is a loose handcuff. A loose handcuff or a tight handcuff in is unbalanced if both its cycles are unbalanced in .
To a signed graph is associated its frame matroid on ground set with rank function defined for by
[TABLE]
When is connected, the bases of correspond to spanning trees of when is balanced, and in that case the two matroids coincide ; otherwise, when is unbalanced, a basis of is the union of edges in a spanning forest of (formed by the trees ), and edges , , such that the unique cycle of in is unbalanced in , for each . In particular, if is connected and balanced, each basis element contains edges, whereas if is connected and unbalanced, it contains edges. For a not necessarily connected signed graph , a basis of is the union of bases of the frame matroids of the connected components of .
The circuits of are the balanced cycles, unbalanced loose handcuffs and unbalanced tight handcuffs.
A loop of is a loop of with positive sign in . A loop of with negative sign in is either contained in no circuit of (when there are no other unbalanced cycles in the same connected component), in which case it is a coloop of , or it belongs to some circuit and is thereby an ordinary edge of . A coloop of (a bridge of ) is also a coloop of , except when it is a circuit path edge (an edge on the path joining the two unbalanced cycles of a loose handcuff), in which case it is ordinary in .
Loops and coloops in signed graphs
The relationships between the notions of loop, coloop and ordinary edge for the matroids and are summarized in Table 2.
3 Tutte polynomials for matroid pairs and for signed graphs
3.1 The Tutte polynomial of a graph and of a matroid
The Tutte polynomial of a graph has subset expansion
[TABLE]
and may alternatively be defined by the recurrence
[TABLE]
and if has no edges. Among its many evaluations with combinatorial interpretations are the following, in which denotes the additive cyclic group on elements:
- •
is the number of proper vertex colorings of with colors,
- •
is the number of nowhere-zero -tensions of (in one-to- correspondence with proper -colorings),
- •
is the number of nowhere-zero -flows of .
The Tutte polynomial of a matroid with ground set and rank function is defined by
[TABLE]
When is the cycle matroid of we have and , so this subset expansion coincides with that given in (2) to define the Tutte polynomial of a graph. The Tutte polynomial of a matroid satisifies mutatis mutandis the same deletion-contraction recurrence given in (3) for the Tutte polynomial of a graph.
3.2 The trivariate Tutte polynomial of a signed graph
Definition 3.1**.**
The trivariate Tutte polynomial of a signed graph with underlying graph is defined by
[TABLE]
It is easy to verify directly that when is switching equivalent to , and that .
The Tutte polynomial of a signed graph can, just as the Tutte polynomial of a graph extends to matroids, be extended to pairs of matroids on a common ground set, as explained in Appendix A. Given a signed graph , let be the cycle matroid of with rank function , and let be the frame matroid of with rank function . Since for each , the polynomial of Definition A.1 is divisible by , as can be seen by inspecting its subset expansion (46). We have then for signed graph with underlying graph ,
[TABLE]
Identities (50) and (51) in Appendix A then imply that the trivariate Tutte polynomial (5) contains as a specialization combinatorial invariants of a signed graph that can be obtained as a specialization of the Tutte polynomial of the underlying frame matroid of or the Tutte polynomial of the cycle matroid of . For example, the number of proper colorings of is an evaluation of the Tutte polynomial of , expressed as an evaluation of the trivariate Tutte polynomial of in Corollary 6.3 below. More generally, the generating function for colorings of according to the number of improperly colored edges or, equivalently, the dichromatic polynomial defined by Zaslavsky [44, Section 3], is given by
[TABLE]
(In a similar way to how the dichromatic polynomial of a graph is up to a prefactor equivalent to the Tutte polynomial of , the dichromatic polynomial of is up to a prefactor the Tutte polynomial of : equation (7) is equation (51) in disguise. Zaslavsky’s dichromatic polynomial is defined more widely for biased graphs, of which signed graphs form a subclass.) By contrast, the number of nowhere-zero flows of in general depends on both and and is not given by an evaluation of the Tutte polynomial of either matroid: to enumerate nowhere-zero flows we require the trivariate Tutte polynomial (see Theorem 5.3 below).
If is balanced, the polynomial coincides with the Tutte polynomial of the underlying graph (as being balanced is hereditary, and so ).
Related work on Tutte polynomials for signed graphs
The Tutte polynomial for signed graphs that we have just defined can be seen as a special case of the huge Tutte polynomial of weighted gain graphs of Forge and Zaslavsky [16], taking all weights equal to and the gain group equal to .
In [24] Kauffman defines a trivariate polynomial for signed graphs that for balanced signed graphs also reduces to the Tutte polynomial of the underlying graph. Godsil and Royle [17, Chapter 15] define a signed rank polynomial for matroids on a signed ground set, specializing to the Kaufmann bracket for links. The polynomial of Definition 3.1 differs from the polynomial of Kauffman since it is invariant under switchings, while the polynomial generally is not. For instance, for the graph in which the edge carries a positive sign the polynomial of Kauffman equals , while for with a negative sign on the edge it equals . These and other signed graph polynomials described in [10, Section 3.2] are specializations of a signed version of the Bollobás-Riordan polynomial [3], which is not invariant under switching.
The trivariate Tutte polynomial may be obtained as a specialization of the “surface Tutte polynomial” of a map (graph embedded in a compact surface), introduced in [20], and whose expression can found in Appendix C, Table 8. An embedding of a graph as a map is commonly represented as a ribbon graph (see e.g. [15]), edges being bands whose two ends are glued along the boundaries of disks representing vertices. With this representation, a sign can be associated with each of the edges of the graph with respect to this embedding, in which an edge receives positive sign when it is untwisted and negative sign if it is twisted. The surface Tutte polynomial of a map contains as a specialization as follows. For an arbitrary embedding of the signed graph into a compact surface (non-orientable precisely when is unbalanced, cf. [34]) as a map ,
[TABLE]
in which and are set equal to the following values: , , for all , if and if .
A variant of the polynomial , in which the exponent of for an in the subgraph expansion is replaced by , appears in the slides of a presentation by Krieger and O’Connor in 2013 [27]. They show that a suitable renormalization of this polynomial equals the Euler characteristic of a chain complex of trigraded modules. This builds upon earlier work on the categorification of, in chronological order, the Jones polynomial by Khovanov [25], the chromatic polynomial by Helme-Guizon and Rong [22] and the Tutte polynomial (for graphs) by Jasso-Hernandez and Rong [23].
4 Deletion-contraction invariants
To define contraction of negative edges in a signed graph requires enlarging the domain of signed graphs by allowing half-arcs and free loops [42]. To avoid doing this, in giving a recurrence for the trivariate Tutte polynomial we only allow contraction of positive edges. As a non-loop can always be made positive by vertex switching, this gives a recurrence terminating in signed graphs consisting solely of bouquets of negative loops.
Theorem 4.1**.**
The trivariate Tutte polynomial of a signed graph with underlying graph satisfies, for a positive edge ,
[TABLE]
and if is the signed graph consisting of negative loops on a single vertex then
[TABLE]
and if has no edges.
Theorem 4.1 can be obtained as a corollary of the framework of [26] applied to the polynomial for matroid pairs, as deletion and contraction of positive edges (and deletion of negative edges) in is compatible with deletion and contraction in the matroids and as usually defined (see [35]). However, we give an independent proof of the recurrence formula for as it is key to many of our results.
Proof of Theorem 4.1.
Given a signed graph with underlying graph and an edge with , we split the subset expansion of into two parts
[TABLE]
according to whether a subset contains , in which case the corresponding term is contained in , or does not contain , in which case the corresponding term is contained in . Let have underlying graph and underlying graph . We have a bijection defined by .
For ,
[TABLE]
and
[TABLE]
Thus
[TABLE]
and
[TABLE]
as . Moreover,
[TABLE]
where we have used that when is a loop in .
Also,
[TABLE]
After these preparations, we may now prove each case of the recurrence, starting with the case of ordinary edges (9). Take an edge not a loop or bridge such that . Then by equations (17), (18) and (14). Equations (19) and (15) imply that , so that .
For case (10), if is a bridge of then by equations and , and by equations and . Thus (13) in this case becomes .
Case (11) follows from (10), for if is a bridge but not a circuit path edge of then . Indeed, for , (since is a bridge of ) and . Also, for every , since in at least one endpoint of is contained in a balanced connected component of (as is not a circuit path edge of ). This means that in there is one less balanced component than in . The same reasoning establishes the equality .
For case (12), assume now that is a loop with . Then by equations (17), (18) and (14), and by equations (19) and (15). As for a positive loop , we have and subsituting into (13) yields .
Finally, assume that is the one-vertex signed graph with negative loops. The contribution of to is . Let be a subset of size . Then
[TABLE]
Hence,
[TABLE]
∎
A balanced signed graph , switching equivalent to a signed graph with all edges positive, can be identified with its underlying graph . In this case, the recurrence for in Theorem 4.1 reduces to that of the Tutte polynomial of as there are no circuit path edges or negative loops.
The Tutte polynomial of a graph as a polynomial in and has non-negative coefficients (evident from its deletion-contraction recurrence, but not so evident from its subset expansion, which involves terms of the form ); furthermore, for a connected graph , the coefficient of in has a combinatorial interpretation as the number of spanning trees of internal activity and external activity . The recurrence of Theorem 4.1 indicates that as a polynomial in and the trivariate Tutte polynomial in general has negative coefficients, as confirmed by the following example.
Example 4.2**.**
Let be the signed graph on two vertices having a unique (positive) bridge and two negative loops, one at each vertex (see Figure 3).
Then, taking in Theorem 4.1, we have , while (by multiplicativity over disjoint unions and taking in Theorem 4.1). By the recurrence (10) we have .
4.1 The Recipe Theorem
The Tutte polynomial is universal for deletion-contraction graph invariants in the sense that if is a graph invariant multiplicative over disjoint unions and satisfying
[TABLE]
and if is a bouquet of loops, then
[TABLE]
(See e.g. [7].) Despite the divisions by and , the formula for also holds for and as the right-hand side upon expanding by the subset expansion for the Tutte polynomial is a polynomial in and .
A similar “Recipe Theorem” holds for the trivariate Tutte polynomial, which we shall apply to obtain combinatorial interpretations of its evaluations at various points in Sections 5–7.
Theorem 4.3** (Recipe Theorem).**
Let be an invariant of signed graphs invariant under switching and multiplicative over disjoint unions. Suppose that there are constants and , with , such that, for a signed graph with underlying graph and positive edge ,
[TABLE]
while if is a bouquet of negative loops then
[TABLE]
and when is a single vertex with no edges.
Then,
[TABLE]
a polynomial in and over .
If or then we use the subset expansion of the right-hand side of (20):
[TABLE]
Proof.
Consider By the recurrence for , we have
[TABLE]
with
[TABLE]
for a bouquet with negative loops and for a single vertex edge-less graph. There is an additional case in the recurrence with respect to Theorem 4.1 for an edge that is ordinary in and : the connected component containing is unbalanced in and balanced in .
Introducing the parameters yields the result.
It is clear from the recurrence formula that is a polynomial in and over . In the given subset expansion (21), all the exponents except possibly that of are nonnegative: for we have , , , and , while may be negative (for example, when is an edge with a negative loop on either endpoint and comprises the two loops). ∎
5 Flows
Flows on signed graphs taking values in an abelian group are defined in a similar way to flows on graphs. Given a graph , we call a pair with and an edge containing a half-edge. (A loop comprises two half-edges.) A bidirected graph is a pair , where is a graph (not necessarily simple) in which every half-edge receives an orientation (The two half-edges associated with a loop at a vertex consist of the same vertex-edge pair but receive orientations independently.) We call the orientation compatible with the signature of a signed graph if for each edge we have
[TABLE]
(In particular, if the sign of a loop is negative then its two half-edges receive the same orientation sign.) A half-edge oriented positively points into , and oriented negatively points out of ; for an edge , when the half-edges and are consistently directed, while if they are oppositely directed. A vertex switch at has the effect of changing the orientation of all the half-edges incident with ; thus the new orientation is compatible with the new signature.
Let be a finite additive abelian group. Considered as a -module, for we have , and . The subgroup will play a significant role in the sequel; the quotient group of cosets is isomorphic to the subgroup (in particular, ).
Flows on bidirected graphs were introduced by Bouchet [6]. A -flow of a bidirected graph is a function such that at each vertex of the Kirchhoff law is satisfied, that is, for each vertex ,
[TABLE]
where the summation runs over half-edges incident with , so if is a positive loop it contributes with two terms to the sum. A -flow of a signed graph is a function such that is a -flow for the bidirected graph , where is an orientation of compatible with . A -flow is nowhere-zero if for all . We let and denote the number of -flows and number of nowhere-zero -flows of , respectively.
Given , and by considering instead of as the value of a flow at , we see that the number of nowhere-zero -flows does not depend on the exact values of and but only on the value of their product. Hence the notion of nowhere-zero flow on a signed graph is well-defined and the number of nowhere-zero -flows is an invariant of signed graphs as it does not depend on the choosen orientation. Furthermore, the number of nowhere-zero -flows is constant on equivalence classes of signed graphs, as switching a vertex reverses the orientation of those half-edges incident with and just replaces the left-hand side of equation (23) with its negation. If a signed graph is balanced, then the number of (nowhere-zero) -flows only depends on the size of the group [37], but if the signed graph is unbalanced, then it also depends on the group structure of , as described by the following theorem.
Theorem 5.1**.**
The number of -flows of a signed graph is equal to
Proof.
The number of -flows of satisfies the recurrence
[TABLE]
since for a non-loop any given -flow of by the defining equations (23) extends to a -flow of by the assignment of a unique value in to , and for a positive loop any value in assigned to contributes zero to equation (23). Furthermore, for a bouquet of loops negative in , as
[TABLE]
The deletion-contraction recurrence for is not of the form given in Theorem 4.3 (which would require taking for it to fit). However, here we can argue directly. Contraction of graph non-loops in leaves a graph consisting solely of graph loops (the nullity of the graph , the dimension of its cycle space). Among these loops, the positive loops in each contribute ; each negative loop in also contributes , but a scale factor of is applied for each bouquet containing a negative loop – the latter correspond to unbalanced connected components of . Hence
[TABLE]
using . ∎
Theorem 5.1 and an application of inclusion-exclusion yields the following subset expansion formula for the number of nowhere-zero -flows.
Theorem 5.2**.**
Let be a finite additive abelian group and a signed graph with underlying graph . Then the number of nowhere-zero -flows of is given by
[TABLE]
Theorem 5.2 is a special case of [20, Theorem ], namely [20, Corollary ]. However, since we deal here with abelian groups, the technical difficulties involved in the proof of [20, Theorem ] when general finite groups are involved can be avoided (as the proof of Theorem 5.1 shows). Very recently, Theorem 5.2 was found independently by Qian [36, Theorem 4.3].
Theorem 5.2 can also be obtained as an evaluation of the trivariate Tutte polynomial by using the deletion-contraction recurrence established in [13] and applying Theorem 4.3, and then using the subset expansion of the trivariate Tutte polynomial as given in Definition 3.1.
Theorem 5.3**.**
Let be a finite additive abelian group. Then, for a signed graph , the number of nowhere-zero -flows of is given by
[TABLE]
Proof.
As shown in [13], for a positive edge (after appropriate switching of vertices, an edge in a signed graph that is not a negative loop can be made positive),
[TABLE]
To see this note that for a positive loop , any value assigned to contributes to the sum (23), from which it follows in this case that . Otherwise, if is not a loop, then for a given nowhere-zero -flow of there is by the defining equations (23) for a flow a unique value we can assign to to extend this flow of to a flow of . Those extensions that take the value [math] on are precisely the nowhere-zero -flows of . This establishes the recurrence.
If is a bouquet of negative loops then, as shown in [13] (simplifying by the binomial expansion the expression given in [13, Lemma 2.1]),
[TABLE]
The expression of as an evaluation of now follows by taking in Theorem 4.3. ∎
When , i.e. is of odd order, the number of nowhere-zero -flows of given in Theorem 5.3 is the evaluation of the Tutte polynomial of the frame matroid . This is a consequence of the identity
[TABLE]
which follows from equation (51) and . When is of even order the number of nowhere-zero -flows of is not an evaluation of the Tutte polynomial of the frame matroid.
Example 5.4**.**
For a graph , the number of nowhere-zero flows of is equal to , a polynomial in . For a signed graph , since when is odd and when is even, the number of nowhere-zero -flows is a quasipolynomial in of period .
6 Colorings
Zaslavsky [42] introduced a notion of signed graph colorings as follows.
Definition 6.1**.**
Let be an integer. A proper -coloring of is an assignment such that for every edge we have . A proper non-zero -coloring is a proper -coloring which does not assign the value [math] to any vertex.
In [33], Máčajová, Raspaud and Škoviera call an -coloring in Zaslavsky’s sense a -coloring, and call a non-zero -coloring in Zaslavsky’s sense a -coloring, with the advantage that the corresponding notion of chromatic number of a signed graph agrees with the (usual) chromatic number of a balanced signed graph (viewed as a graph). We will however use Zaslavsky’s terminology.
Let be a signed graph. For , define to be the number of proper -colorings and to be the number of proper non-zero -colorings of . Zaslavsky showed that these are both polynomials in , and established subgraph expansions for them as follows.
Theorem 6.2** (Theorem in [41]).**
Let be a signed graph. Then
[TABLE]
and
[TABLE]
Zaslavsky further showed that these chromatic polynomials evaluated at negative integers have interpretations similar to the chromatic polynomial of a graph evaluated at negative integers in terms of colorings and compatible orientation [41, Theorem 3.5].
Theorem 6.2 and the subgraph expansion of the trivariate Tutte polynomial (Definition 3.1) immediately yield an expression for the number of proper (non-zero) -colorings as an evaluation of the trivariate Tutte polynomial.
Corollary 6.3**.**
Let be a signed graph. Then the number of proper -colorings of is given by
[TABLE]
and the number of proper non-zero -colorings is given by
[TABLE]
The number of proper -colorings is an evaluation of the Tutte polynomial of the frame matroid at , i.e. similarly to how the number of proper -colorings of a graph is the evaluation . See Corollary 6.6 below. The number of proper non-zero -colorings, however, is not an evaluation of the Tutte polynomial of . For example, while a balanced complete graph on three vertices and the unbalanced loose handcuff on two vertices (Figure 3) have the same frame matroid (in either case, any pair of edges forms a basis), a balanced complete graph has no non-zero -colorings while the unbalanced loose handcuff has two.
Zaslavsky’s notion of proper (non-zero) -colorings of signed graphs may be generalized as follows to colorings taking values in a finite set equipped with an involution on .
Definition 6.4**.**
A proper -coloring of a signed graph with vertices is a map such that, for an edge , we have if is positive and if is negative.
If is an additive abelian group of order and if is the involution , for , then the definition of a proper -coloring is equivalent with Zaslavsky’s definition of a proper -coloring.
Let denote the number of proper -colorings of . The following theorem establishes that is an evaluation of the trivariate Tutte polynomial similar in form to the specialization of the Tutte polynomial of a graph to the classical chromatic polynomial,
[TABLE]
counting the number of proper vertex colorings of using a finite color set .
Theorem 6.5**.**
For a signed graph with underlying graph , the number of proper -colorings of is given by
[TABLE]
where .
Proof.
For a map , define (the set of impropriety of , to use Zaslavsky’s term) by
[TABLE]
We also define
[TABLE]
the number of colorings of improper on every edge. Then
[TABLE]
To see why equation (29) holds, we may assume that is connected. A map for which then is uniquely determined by the value it assigns to a fixed vertex of . If is balanced, there are choices for this value. If is unbalanced, then the presence of an unbalanced cycle forces this value to satisfy , yielding choices for .
From equation (28) it follows that for we have
[TABLE]
Using inclusion-exclusion (in the third equality below), we then calculate that
[TABLE]
The results follows upon substituting and into the subset expansion formula (5) defining . ∎
Alternatively, Theorem 6.5 can be shown by establishing that the number of proper -colorings satisfies the deletion-contraction of Theorem 4.3 with ; an additional prefactor of arises as the number of proper -colorings for a single vertex is .
The focus of Section 7 will be proper -colorings for a finite additive abelian group , by which we mean proper -colorings in which is the set of elements of and is negation (additive inverse). The number of fixed points of in this case is . Let denote the number of proper -colorings. By Theorem 6.5 we then have the following:
Corollary 6.6**.**
Let be a finite additive abelian group. Then, for a signed graph , the number of proper -colorings is given by
[TABLE]
7 Tensions and potential differences
For a finite additive abelian group , there is a correspondence between -tensions of a graph and colorings of the vertices of by elements of : for each such coloring, taking the difference between endpoint colors (the order the difference is taken according to a fixed orientation of ) yields a unique -tension, which is nowhere-zero precisely when the coloring is proper. Conversely, to each -tension there correspond vertex colorings of . Therefore tensions of a graph coincide with “potential differences” of vertex colorings (see e.g. [2]).
An analogous notion of a -tension [9] exists for signed graphs; only, unlike for graphs, not every -tension arises from a vertex coloring. For signed graphs, then, the notion of a potential difference does not coincide with the notion of a tension established in [9]: potential differences of a -coloring of an unbalanced signed graph form a proper subset of -tensions unless is of odd order.
We shall define tensions in a different but equivalent way to [9], and introduce -potential differences as -tensions with an added constraint (the relationship between -potential differences to -colorings will emerge in Section 7.2). Before giving a formal definition, we require some preliminaries.
A walk of a signed graph with underlying graph , written as a vertex-edge sequence
[TABLE]
is said to be positive in if and negative otherwise. The walk is a closed walk if . Recall from Section 2.2.2 that a circuit of a signed graph is an edge set forming a balanced cycle, or an edge set forming two unbalanced cycles sharing exactly one common vertex (an unbalanced tight handcuff), or an edge set forming two vertex-disjoint unbalanced cycles joined by a simple path meeting the cycles exactly in its endpoints (an unbalanced loose handcuff). In the last case, the edges of the path are called circuit path edges; otherwise edges of a circuit belong to a cycle of the graph . Circuits naturally give rise to positive closed walks in the signed graph with circuit paths edges being used twice and other edges once. We call such closed walks circuit walks.
Definition 7.1**.**
Let be a signed graph, let be an orientation compatible with and let be a finite additive abelian group. A map is a -tension of with respect to the orientation if and only if, for each circuit walk ,
[TABLE]
The map is said to be a -potential difference of if and only if is a -tension such that, for every walk around an unbalanced cycle,
[TABLE]
See Figure 4 for an example of (30). When has odd order, equation (31) is always satisfied, so that -potential differences coincide with -tensions in this case.
Remark 7.2*.*
Our definition of tension is equivalent to the definition of tensions in [9, (4.4)], as (in the notation of [9]) is equal to when the orientation of the walk has the half-edge going out of . See Appendix B for more details.
Remark 7.3*.*
Although Definition 7.1 depends on the choice of orientation of compatible with and involves a choice of starting vertex for the walk , the number of -tensions of is independent of both choices: changing the direction of an edge in corresponds to negating the value of each -tension on ; starting the walk at a different vertex, say , has the effect of multiplying equation (30) by .
Switching at a vertex in flips the sign function on edges incident with , making an equivalent signed graph . Orientations of compatible with are transformed into orientations of compatible with by switching the sign of on each half-edge incident with . The proof of the following proposition consists in showing that the equations (30) for circuit walks that define a tension of are preserved under vertex switching.
Proposition 7.4**.**
The set of -tensions and the set of -potential differences of a signed graph are invariant under switching.
Proof.
We just need to show that the set of -tensions is invariant under switching, as the condition (31) for a -potential difference is independent of edge signing and orientation.
The effect on of switching at a vertex is to multiply its value on each half-edge incident with by ; the signs of edges incident with are likewise flipped (except for loops on : both half-edges of a loop at are multiplied by , thus preserving the sign of the loop).
By Remark 7.3, it suffices to consider circuit walks starting and finishing at a given vertex . The coefficient of in equation (30) has its sign flipped. We now describe the effect on the coefficients of for . As is a circuit walk, there are just two cases to consider: there is no such that , or there is exactly one such that .
Consider the first case where for , and the walk does not simply traverse a loop on . Then, for each , the coefficient of in equation (30) has its sign flipped, as is the only sign that is changed (along with , which does not feature). Therefore, after switching at , equation (30) is replaced by its negation, thus preserving the solutions for to the equation. When there is a loop on , the only possibility in this case is the walk , for which the coefficient of in equation (30) is equal to , which is flipped in sign under switching at .
Consider now the second case where for exactly one , i.e., . First we assume , i.e. neither nor are loops on . The same argument as for the first case shows that the sign of the coefficient of is flipped for each . The coefficient of has its sign flipped, as , and are negated and the remaining signs are preserved. For the coefficient of in equation (30) has its sign flipped as , and are negated and the remaining signs are preserved. Again, the overall effect is to negate the whole of equation (30).
When there is a loop on , we may assume , where is not a loop if (so the underlying circuit is an unbalanced loose handcuff), while if then where is another loop on (so the underlying circuit is an unbalanced tight handcuff). For the case of a loose handcuff, the coefficient of is , which is flipped in sign under switching at . The coefficient of is , in which has signed flipped under switching at (while the sign of is preserved since is a loop). For , the coefficient of is , which is flipped in sign as has its sign flipped. For the case of the tight handcuff , both coefficients and are flipped in sign (the sign of is preserved while both and have flipped signs). Hence again the overall effect is to negate the whole of equation (30). ∎
Proposition 7.4 implies that if is balanced, and hence switching equivalent to an all-positive signed graph, then a -tension of corresponds exactly with a -tension of its underlying graph (see, for instance, “global -tension” in the terminology of [18, 32]). It follows that if is balanced then every -tension of is a -potential difference.
Tensions are equivalently defined by stipulating that the net sum around any positive closed walk (not just a circuit walk) is zero, as expressed by the following proposition.
Proposition 7.5**.**
With the same notation as in Definition 7.1.
- •
The map is a -tension if and only if (30) is satisfied for every positive closed walk .
- •
The map is a -potential difference if and only if is a tension and (31) is satisfied for every negative closed walk .
Proof.
Let be a finite abelian group, a signed graph, an orientation compatible with , and a -tension of , i.e. satisfying equation (30) for each circuit walk . In order to show that equation (30) is satisfied for every positive closed walk , we argue by contradiction, showing that any counterexample must contain a subwalk of smaller nonzero length that is also a counterexample, or implies that a circuit walk is also a counterexample. As neither option is possible, we conclude that no counterexample exists.
We introduce some terminology and notation needed in the proof. The concatenation of walks and is defined by . A closed walk can be expressed as a concatenation of two walks and in which starts at an arbitrary vertex of the walk , and starts at the last vertex of and finishes at the first vertex of . Given a walk , its length is , and its interior vertices are the vertices (which may include the first or last vertex if revisited by the walk). A walk is nontrivial if it has non-zero length, and a subwalk of a walk is proper is it is neither trivial nor the whole walk . Let , and, for , let . The height of on is then defined as
[TABLE]
Definition 7.1 says that is a -tension if and only if for all circuit walks . Our goal is to show that for any positive closed walk.
We begin by collecting some properties of the height function that we shall use. For , a concatenation of walks,
[TABLE]
If is a positive closed walk then we have , and thus
[TABLE]
We may therefore cyclically permute a positive closed walk to bring any of its subwalks as its initial subwalk without affecting the property that . (This extends the observation made in Remark 7.3 that the defining equations (30) for a -tension do not depend on the starting vertex chosen for each circuit walk .) The reverse walk of a walk is the walk We have and more generally for . We observe that
[TABLE]
where in the first line we use that by compatibility of with . As a consequence, for walks and for which the concatenation is defined,
[TABLE]
[TABLE]
Assume now that is a positive closed walk of minimum nonzero length such that . We establish a series of claims which together imply that cannot exist.
If for nontrivial closed walks and then both and must be negative for otherwise by (32) and one of and is of smaller length than . Thus,
[TABLE]
By the number of times a vertex occurs in a closed walk we mean the number of indices such that .
[TABLE]
To see this, suppose for nontrivial closed walks each starting and ending at vertex . (By (33) we may assume starts at .) As is positive, not all of can be negative. By applying (33) if necessary, we may take to be positive. By (32) we have . Since , either or . But and are positive closed walks of smaller length than . This contradicts minimality of and establishes (37).
We say that a nontrivial subwalk of closed walk appears more than once in if (starting at the first vertex of in one of its appearances as a subwalk).
[TABLE]
Suppose to the contrary that . The walk is closed as the last vertex of is the first vertex of , and hence by (36) cannot be positive. Since is closed and positive, the walk is closed and positive. By (32) we have, since ,
[TABLE]
where the last equality follows from (34). This implies that . But is a positive closed walk of smaller length than , contradicting minimality of . This establishes (38).
[TABLE]
Suppose on the contrary that for proper closed subwalk and subwalks such that is an interior vertex of that appears again as the last vertex of and first vertex of . Let where ends in and begins with ( and are nontrivial as is an interior vertex of ). By (36), must be negative, and then is also negative. Thus and have opposite signs and and have opposite signs. Since and are closed subwalks of too, they are negative by (36). This forces . Since and are positive closed walks of smaller length than we have
[TABLE]
and similarly . But, using and ,
[TABLE]
a contradiction. This establishes (39).
[TABLE]
To see this, suppose on the contrary that for nontrivial closed subwalks and walks (we may assume begins with one of its closed subwalks by (33), and by (39) two proper closed subwalks share no interior vertices). By minimality of , each of and must be negative. By using (33) if necessary, we may assume has minimum length among . Let and . The walk is closed (since starts at the end of and ends at the start of ), positive (since and ), and shorter than (since has strictly smaller length than by assumption on ). Thus is also closed, positive, and of smaller length than . By (35) and (32), , which leads to the contradiction that one of the shorter closed positive walks and has nonzero height. This establishes (40).
Properties (36), (37), (38), (39) and (40) together imply that for simple closed negative walks and and paths and such that is a positive closed walk, and, by minimality of , . We have
[TABLE]
and since is a circuit walk, we must then by (30) have . This final contradiction establishes that there is no counterexample to the first part of the proposition.
To prove the second part of the proposition characterizing -potential differences, we begin with the observation that for an arbitrary walk the height belongs to the same coset of as , since for any . Thus the constraint (31) defining a -potential difference is equivalent to for every walk around an unbalanced cycle. A negative closed walk must pass through a vertex belonging to a walk traversing an unbalanced cycle. (By vertex switching we may assume all edges not in cycles traversed by are positive; then there must be a negative cycle that meets.) Making the start and end vertex of (permitted by (33)), the walk is positive and hence . By (31), , whence , and the result follows. ∎
The set of -tensions of a connected signed graph can be partitioned into subsets according to the height of walks traversing unbalanced cycles modulo the subgroup (zero for -potential differences as per equation (31)):
Proposition 7.6**.**
If is a connected unbalanced signed graph and is a -tension, then there is such that for every closed walk around an unbalanced cycle of
[TABLE]
Proof.
We use notation introduced at the beginning of the proof of Proposition 7.5. If and are walks traversing two unbalanced cycles in the same connected component of and a path from the end of to the start of , then the concatenation is a positive closed walk. By Proposition 7.5, the -tension then satisfies (equation (30)). Using (34)
[TABLE]
which, as for any , implies
[TABLE]
This in turn implies belongs to the same coset of as . ∎
7.1 Enumeration of (nowhere-zero) -tensions and (nowhere-zero) -potential differences
We let , , and denote the number of -tensions, nowhere-zero -tensions, -potential differences and nowhere-zero -potential differences, respectively.
Much as a -tension of a connected graph is uniquely determined by its values on a spanning tree of , so that there are clearly -tensions of , an analogous statement holds for -tensions and -potential differences of a connected signed graph, proved as Theorem 7.9 below, with the notion of a connected basis of a signed graph replacing that of spanning tree of a graph.
Definition 7.7**.**
Let be a connected signed graph. A connected basis of is a spanning tree of when is balanced, and, when is unbalanced, the union of a spanning tree of and an edge such that the unique cycle in the subgraph is unbalanced in . A connected basis of the disjoint union of connected signed graphs is the union of connected bases of the connected components.
For a balanced connected signed graph, all bases (maximal subsets of edges containing no circuit) are connected so the qualifier “connected” in “connected basis” is redundant in this case. However, for unbalanced signed graphs, maximal subsets of edges containing no circuit may be disconnected. In fact, any forest of connected bases of unbalanced induced subgraphs of that together cover all the vertices of forms a basis as the addition of any edge forms either a balanced cycle or a handcuff.
Lemma 7.8**.**
A -tension of a connected signed graph is uniquely determined by its values on the edges of a connected basis (any choice of values on the basis determines a unique tension on the whole of ).
Lemma 7.8 can be proven using results in [9]; we discuss this connection and give an alternative proof in Appendix B.
Theorem 7.9**.**
The number of -tensions of a signed graph is given by
[TABLE]
The number of -potential differences is given by
[TABLE]
Proof.
By multiplicativity of the number of -tensions and -potential differences of a signed graph over disjoint unions, it suffices to prove that for a connected signed graph on vertex set the number of -tensions is given by
[TABLE]
and the number of -potential differences is given by
[TABLE]
By Lemma 7.8, there is a bijection between -tensions of and -valued functions on a connected basis. If a connected signed graph with underlying graph is balanced, then a basis has edges, while if it is unbalanced, then a basis has edges. The additional coset condition implies the factor in the -potential differences instead of the factor for the -tensions since a basis contains an unbalanced cycle . While Lemma 7.8 does not directly give that this tension will automatically be a potential difference, it does follow from it. Indeed if there is another unbalanced cycle in then we can traverse and by walks and appeal to Proposition 7.6 to show that the sum of the tension values on the two unbalanced cycles belongs to . ∎
Although Theorem 7.9 gives a formula for the number of -tensions, the usual route of inclusion-exclusion in order to obtain a subset expansion for the number of nowhere-zero -tensions is not available to us as contraction of negative edges has not been defined for signed graphs.111Recall that for a graph the number of -tensions of is equal to so that by inclusion-exclusion the number of nowhere-zero -tensions of is
Here we use the fact that and that a tension of corresponds to a unique tension of that is zero on (dual to the fact that a flow of corresponds to a unique flow of that is zero on ). For nowhere -flows of signed graphs we were able to use inclusion-exclusion – see Theorem 5.1; for proper -colorings too, we relied on the inclusion-exclusion formula used to derive Theorem 6.5 to obtain the number of proper -colorings 6.6 as an evaluation of the trivariate Tutte polynomial. (Once we have explained the connection between colorings and tensions, we shall see that this in fact gives the number of nowhere-zero -potential differences – see Corollary 7.15 below.)
In order to enumerate nowhere-zero -tensions we shall establish a deletion-contraction recurrence, but in order to do so we need to partition tensions into classes. Let denote the number of nowhere-zero -tensions such that for every walk around an unbalanced cycle of
[TABLE]
By Proposition 7.6, for a connected signed graph each nowhere-zero -tension falls into one of these classes, when , while (restricting the range of in the sum to a transversal of cosets of gives ).
Theorem 7.10**.**
Let be a signed graph and a finite additive abelian group. Then the number of nowhere-zero -potential differences of is given by
[TABLE]
and, for ,
[TABLE]
In particular, when is connected and unbalanced, the number of nowhere-zero -tensions of is given by
[TABLE]
Remark 7.11*.*
When is balanced, -tensions of are -potential differences of and correspond to -tensions of ; when is balanced when and (as can be verified from the subset expansion for the trivariate Tutte polynomial). So when is balanced.
Proof.
We claim that for each the invariant satisfies the following deletion-contraction recurrence: For positive edge ,
[TABLE]
and for a vertex with negative loops we have
[TABLE]
Then the result will follow by Theorem 4.3 with .
It suffices to consider connected . We take the cases in turn, letting be a nowhere-zero -tension of with sum in on unbalanced cycles, and counting how many ways can be extended to a nowhere-zero -tension of also with sum in on unbalanced cycles.
When is ordinary in and in , the edge either appears in some balanced cycle (a circuit of both and ) or in an unbalanced cycle of a handcuff (the cycle is a circuit of and the handcuff, loose or tight, is a circuit of ). The value of is by Lemma 7.8 uniquely determined by the partial tension on . Those tensions for which correspond to tensions of . Hence in this case.
When is ordinary in and it belongs to each unbalanced cycle of (of which there is at least one). For such an unbalanced cycle , the condition determines possible values of ; choices correspond to tensions of . These conditions are compatible for all unbalanced cycles as two such cycles together form a positive closed walk. Since there are no unbalanced cycles in , the only circuits of are balanced cycles. The positive edge cannot both belong to an unbalanced cycle and to a balanced cycle, for otherwise there would be an unbalanced cycle not containing . Therefore there are no further constraints on beyond the sum of values on the cycle being in . Subtracting for when the choice is available (corresponding to a tension of ), we thus obtain in this case.
When is a bridge of and circuit path edge, then there is a circuit of containing and equation (30) for a circuit walk around determines , and – this is where the condition that summed on each unbalanced cycle belongs to a fixed coset is required – this makes choices for . (If two unbalanced cycles contained in a loose handcuff had incongruent sums modulo then equation (30) could not be satisfied for the circuit walk around this loose handcuff.) Adjusting again for the possibility , this yields in this case.
When is a bridge of that is not a circuit path edge, there are no circuits of containing and thus there are non-zero choices for to extend the tension of to a tension of (all the while keeping the sum on any unbalanced cycles an element of ). Thus in this case.
When is a positive loop it forms a circuit of and the equation (30) for a circuit walk around it determines that . Hence in this case.
Finally, when consists of a vertex with negative loops the value on one loop determines all the others to have the same value by equation (30) for circuit walks around pairs of loops. The value on a loop must also belong to . This implies when and when .
This completes the proof of the recurrence and the Recipe Theorem yields the result. ∎
Remark 7.12*.*
For graphs, (nowhere-zero) -tensions are dual to (nowhere-zero) -flows in the sense that tensions of the oriented cycle matroid of are flows of the dual oriented cycle matroid (isomorphic to for planar ). This is reflected in the fact that the number of nowhere-zero -tensions is equal to and the number of nowhere-zero -flows is equal to , which by the duality formulafor the Tutte polynomial is equal to .
In a similar way, the number of nowhere-zero -potential differences given in Theorem 7.10 is equal to
[TABLE]
where and , and the number of nowhere-zero -flows of is equal to
[TABLE]
the last line by the duality formula for given by Equation (53) in Appendix A.
What remains unclear, though, is how to define -flows and -potential differences for the dual of frame matroids. Tensions and flows of (the cycle matroid of) a graph have a smooth translation to cographic matroids; but for frame matroids it is not apparent, for example, how the condition on -potential differences that sums on unbalanced cycles belong to translates to the dual setting.
While for connected is a sum of evaluations of the trivariate Tutte polynomial at two different points, and is multiplicative over disjoint unions, the invariant does not itself satisfy a deletion-contraction recurrence. To see this, it suffices to consider the case of edge being a bridge in that belongs to a loose handcuff of . We introduce notation for a number of signed graphs: , consisting of a single vertex with one negative loop, and a single vertex with two negative loops; , consisting of two vertices connected by an edge with one negative loop on each vertex, and two vertices connected by an edge, one negative loop on one, and two negative loops on the other; , consisting of a path on three vertices with a negative loop on each vertex. Supposing there exist and (possibly depending on ) such that , then we have
[TABLE]
and
[TABLE]
From these two equations it follows that
[TABLE]
but a direct computation shows
[TABLE]
and for and (for example) equality (42) does not hold.
7.2 Tensions and colorings
We describe a relation between tensions and colorings, thereby extending Theorem 5.1 in [9] (where only abelian groups of odd order are considered). As usual, is a finite additive abelian group, and is a signed graph whose underlying graph has been given an orientation compatible with . Define the difference operator with respect to the orientation (sometimes we will make the dependence explicit by writing ) for by
[TABLE]
Changing the orientation of on an edge has the effect of replacing by . The image of the operator is unchanged under vertex switching, and the kernel of is unchanged up to isomorphism. To see this, it suffices to consider the effect of switching at a single vertex to make equivalent signed graph . Under switching at , an orientation of compatible with may be obtained by changing the sign of the half-edges incident with , i.e. by setting for each edge with as an endpoint, and otherwise. For given , if we define by while setting for , then, for each edge incident with ,
[TABLE]
For edges not incident with it is clear that . Hence the image of under the orientation of is equal to the image of under the orientation of ; likewise, under the given orientation of if and only if under the orientation of , and is an isomorphism of .
Theorem 7.13**.**
Let be a finite additive abelian group and a signed graph with underlying graph . Then is a group homomorphism under pointwise addition, whose image is contained in the group of -tensions of , and
[TABLE]
where .
Proof.
It is not difficult to see that is a group homomorphism and that is a -tension for every (see also the first part of the proof of Theorem 5.1 in [9], which holds for any finite abelian group ).
We now determine the kernel of . It suffices to prove that, when is connected,
[TABLE]
By Lemma 7.8, the value of a tension is determined by its values on a connected basis.
Suppose that is balanced. Then, a connected basis is a spannning tree. Let be such a tree. Let . Consider a vertex . Let be an edge in with . Then the vertex receives the value by (43). If , then receives the value . In general a vertex receives the value or depending on the parity of the number of negative edges on the unique path in from to . Thus, each depends on the value on a given vertex, and such value can always be freely chosen, and the values on the other vertices is propagated through a spanning tree. As the component is balanced, the assignment is consistent.
It remains to prove (44) when is unbalanced.
In one direction, take and for define by . Then , showing that contains a subgroup isomorphic to .
In the other direction, let be a connected basis of , in which is a spanning tree of the underlying graph . By Lemma 7.8, we know that a tension is uniquely determined by its values on . Assume, without loss of generality, that (take to be an edge signed in the unique unbalanced cycle of ). Take and consider the value of on a vertex . As in the balanced case, if , and , then , while if , then , due to (43) and the assumption . In particular, the value of at each vertex is either or depending on the number of edges signed on the unique path in from to . Now consider the unique unbalanced cycle containing . Since and the cycle is unbalanced, the path from to has an even number of edges signed . Thus . Equation (43) then states that . Therefore, , and the same is true of , and we are done. ∎
Equation (43) implies that if is a -coloring of , then is a nowhere-zero -tension of .
For a balanced signed graph , the difference operator is surjective ([9, Theorem 5.1]). If is unbalanced, the image of has the following characterization.
Theorem 7.14**.**
Let be an unbalanced signed graph. Let be a -tension of . Then is in the image of if and only if for each unbalanced cycle of
[TABLE]
Thus, is in the image of if and only if it is a -potential difference.
Proof.
Suppose first that for some . The terms of the sum
[TABLE]
contain for each two appearances of with coefficient , and therefore in total has coefficient in , and the overall sum is a multiple of . This proves that if is in the image of , then the sum of its values around an unbalanced cycle is a multiple of in .
Let us now prove the converse. For balanced connected components the statement follows from ([9, Theorem 5.1]). Consider an unbalanced connected component and let be an unbalanced cycle in this component. Let be a connected basis for containing as its unbalanced cycle. By vertex switching we may assume that there is a single edge of that is negative, and that such an edge belongs to , that is, for each and has and .
Traversing as a cycle of in either direction gives a cyclic order on : fix one of these. Set the orientation of compatible with so that for in which follows when traversing , and .
For the given -tension of we have by assumption such that . We construct such that . To start the vertex coloring, we set . We then use the tension to give each vertex of a color as follows. For , with following , we set ; after this, for , with following , and colored, we set . This coloring rule is consistent since . To color the remaining vertices, we use the fact that each vertex in is connected to by a unique path in ; directing these paths outwards from , for each edge with following , colored, we set . Then for , and .
The vertex coloring then has on the edges of ; by Lemma 7.8 the (extended) tension coincides with on the whole of . The procedure is now repeated for each unbalanced connected component and thus a coloring for the whole is found. ∎
Using Theorem 7.14 and Theorem 7.13, from which the number of proper -colorings is given by , we obtain from Corollary 6.6 an alternative derivation for the number of nowhere-zero -potential differences to that obtained by deletion-contraction in Theorem 7.10.
Corollary 7.15**.**
Let be a signed graph and a finite additive abelian group. Then the number of nowhere-zero -potential differences is given by
[TABLE]
Acknowledgements
We thank Sergei Chmutov for alerting us to [16] and [10], Clément Dupont for elucidating to us the relevant parts of [14], Steve Noble for pointing us to [39], Lex Schrijver for leading us to Theorem 6.5, Thomas Zaslavsky for comments on a preprint version of this paper, and the two anonymous referees for their comments on the submitted manuscript.
Appendices
Appendix A The trivariate Tutte polynomial of a pair of matroids on a common ground set
Definition A.1**.**
For matroids and on a common ground set , The trivariate Tutte polynomial of the ordered pair is defined by
[TABLE]
Given a signed graph , let be the cycle matroid of with rank function , and let be the frame matroid of with rank function . Since for each , we see from the subset expansion (46) that the polynomial is divisible by . The trivariate Tutte polynomial of a signed graph with underlying graph , as defined in Definition 3.1, is given by
[TABLE]
In order to establish properties of we shall use an auxiliary four-variable polynomial invariant of matroid pairs first introduced by Welsh and Kayibi.
Definition A.2** ([39]).**
Let and be two matroids on the same ground set. The linking polynomial of the matroid pair is defined by
[TABLE]
The linking polynomial of a matroid pair is in fact a specialization of the trivariate Tutte polynomial as there are just three independent parameters in the exponents – size, rank in , and rank in :
[TABLE]
The trivariate Tutte polynomial in the form of the four-variable linking polynomial is evidently the canonical Tutte polynomial of matroid pairs in the sense of [14]: in the language of that paper, the universal norm of a pair of matroids on a common ground set is
[TABLE]
and the universal Tutte character (in eight variables) boils down by homogeneity to the four-variable linking polynomial.
By setting one of the variables to be constant in equation (49) and relabelling variables we obtain the trivariate Tutte polynomial as a specialization of the linking polynomial. For example, setting in turn and to equal we obtain:
[TABLE]
[TABLE]
Three easily verified properties of the linking polynomial yield the corresponding properties for the trivariate Tutte polynomial. Firstly, the Tutte polynomials of and of are the bivariate specializations
[TABLE]
Secondly, swapping matroids gives
[TABLE]
Lastly, taking dual matroids gives
[TABLE]
(The dual of a matroid is the matroid with rank function , for . The dual matroid is alternatively specified by its bases being the complements of the bases of . The independent sets of are those sets such that there is a basis of contained in . If is the cycle matroid of a planar graph , then is the cycle matroid of the (surface) dual planar graph .)
We may use these properties of the linking polynomial to first obtain that the trivariate Tutte polynomial includes the Tutte polynomial of and the Tutte polynomial of as specializations:
[TABLE]
and
[TABLE]
where the right-hand side of either identity is well-defined for (respectively ) once expanded as a polynomial in (respectively ). The polynomial thus includes among its evaluations enumerations of combinatorial objects associated with or with taken separately, such as the number of connected spanning sets of (evaluating at ).
Secondly,
[TABLE]
Lastly,
[TABLE]
The combination of swapping matroids and taking duals gives
[TABLE]
As noted by Welsh and Kayibi [39], when and form a matroid perspective , which is to say that every circuit of is a union of circuits of , the linking polynomial of a pair of matroids on a common ground set, and hence the trivariate Tutte polynomial of , coincides with the Las Vergnas polynomial. Explicitly, if is a matroid perspective then
[TABLE]
where is the Tutte polynomial of , as given in [31, Equation (5.2)] and first defined by Las Vergnas in [29] (for perspectives of arbitrary length). Every -matroid (see [11] for a definition and extensive treatment of -matroids) gives rise to a matroid perspective between the upper and lower matroid of . Hence, equation (54) also relates the polynomial to the Las Vergnas polynomial of (see Definition 6.1 in [11]). In turn, the Las Vergnas polynomial of the -matroid , where is an embedded graph, specializes to the Las Vergnas polynomial of as defined in [29], since in that case is the cycle matroid of , and is the bond matroid of the surface dual of (Corollary 5.4 in [11]).
Deletion-contraction and the Recipe Theorem for matroid pairs
A deletion-contraction recurrence formula for the trivariate Tutte polynomial follows directly from that given for the linking polynomial in [39, Theorem 1].
Theorem A.3**.**
Let and be matroids on a set , and let . Then the trivariate Tutte polynomial satisfies the following recurrence:
[TABLE]
The invariant
[TABLE]
is easily checked from the subset expansion (46) for to be a polynomial in and (by using for in a matroid on ground set ). By considering the effect of edge deletion and contraction on the prefactor of in (55) and multiplying the recurrence for given in Theorem A.3 by this prefactor, and taking and , we find that this scaled version of the trivariate Tutte polynomial satisfies a recurrence giving a “Recipe Theorem” for deletion-contraction invariants of matroid pairs.
Theorem A.4**.**
Let be an invariant of matroid pairs , in which and , which is multiplicative over direct sums and satisfies the following deletion-contraction recurrence:
[TABLE]
and if .
Then
[TABLE]
Theorem A.4 can be derived from the “Recipe Theorem” [39, Theorem 5] for the four-variable linking polynomial ; the parameters
[TABLE]
in [39] are the parameters
[TABLE]
in our notation. Subject to the condition among the parameters (56), Welsh and Kayibi further show that if an invariant of matroid pairs on a common ground set satisfies a general deletion-contraction formula according to the nine possible edge types, as itemized in [39, Equations (1) to (13)] and featuring the parameters (56), then the dependencies among the parameters (56) are those seen among the coefficients in the recurrence of Theorem A.4. When among the parameters (56) there is “a multitude of very special cases”[39], in each case the invariant taking the form of a single monomial (see [39, Section 7.5] for an example).
Theorem 4.3 (the “Recipe Theorem” for deletion-contraction invariants of signed graphs up to vertex-switching) may be obtained as a corollary of Theorem A.4 upon taking , , discarding the edge type combinations of a loop in and coloop or ordinary edge in as these cannot occur for this particular choice of matroids, and verifying that the reduction of a bouquet of negative loops by deletion and contraction of edges leads to the boundary condition for bouquets of negative loops given in the statement of Theorem 4.3. (A negative loop is a loop in that is ordinary in if there is more than one negative loop in the bouquet and a coloop in if there is just one negative loop. Deletion and contraction of a negative loop is defined for signed graphs in Zaslavsky [42] and requires enlarging the domain of signed graphs to include half-arcs and free loops.)
Appendix B Proof of Lemma 7.8
In this appendix we prove Lemma 7.8, which involves a detailed case analysis; after giving this proof, we then explain how the same result may be derived in the linear algebra framework of [9].
Proof of Lemma 7.8.
We begin by establishing uniqueness. A -tension of a signed graph gives by restriction a -valued function on the edges of a connected basis of . Thus it suffices to prove that any function can be uniquely extended to a -tension of . Consider two tensions, and , that coincide on . Then is also a tension and is zero on . For each , there is a unique circuit walk in consisting of the edges and the signature restricted to these edges. The edge is not a circuit path edge of this circuit walk as is a connected basis. Hence, given the values of on , the tension condition in equation (30) uniquely determines the value on of the extension of to the tension of , and we see that as is identically zero on . Thus for each . This finishes the proof of uniqueness.
The procedure described above gives, for an arbitrary function and edge , a unique assignment determined by a walk around the circuit contained in . It remains to see that this procedure gives a tension, regardless of the basis and the function .
For this purpose, consider the connected graph with the connected bases as vertices, two connected bases joined by an edge if there is an edge exchange transforming one into the other. Every circuit in a connected signed graph can be obtained as a subgraph after adding a unique edge to some connected basis. For a connected basis , let be the set of circuit walks that arise by adding an edge to . By construction, is defined in such a way that the equations (30) for circuit walks from are satisfied. The equations (30) given by a circuit walk in for any connected basis are all satisfied by provided the following claim holds:
If edges and are added to a generic connected basis , and the value is determined by equation (30) for the unique circuit walk in , , then equation (30) is satisfied by for each circuit walk in .
Indeed, for each change of connected basis , when extending to the value determined by equation (30) for a walk around the unique circuit in is such that, when extending to , the value determined by equation (30) for a walk around the unique circuit in coincides with the value . Then, for each connected basis , not only are the equations given by circuit walks in satisfied, but also those in .
The above claim is established by a case analysis detailed in Figures 5,6,7,8,9 and associated Tables 3,4,5,6,7 to see that all the equations given by circuit walks in are determined by those given by the circuit walk in and the circuit walk in . This finishes the proof of Lemma 7.8. ∎
Explanation of the figures.
In the tables and figures given below, the relevant cases occur when the signed graph is connected and unbalanced; however, also included are the balanced cases (remove an edge within one of the cycles that is forced to be negative to turn an unbalanced connected basis into a balanced connected basis; in balanced cases we may furthermore assume that all the signs of the edges are positive, and the coefficients in the given equations become [math]). The various cases of the case analysis are obtained by considering a generic tree with an additional edge closing a negative cycle (to make a connected basis of an unbalanced signed graph), adding an edge , and then adding an additional edge : for instance, Figure 5 arises by considering that the first edge closes another cycle, independent of the basis cycle, and then is a chord of one of the cycles. All the figures of type I are those in which the first edge closes a cycle independent of the basis cycle (that is, it is not a chord of the negative cycle of the given connected basis), while the figures of type II are those in which the first edge is a chord of the basis cycle. Some of these figures can be obtained in different ways.
Leaves are not relevant in equations (30); therefore Figures 5,6, 7, 8 and 9 only give the different schematic structure of the circuits when two edges are added to a connected basis of an unbalanced signed graph (which contains a negative cycle). In order for Figures 5,6, 7, 8 and 9 to be exhaustive, some paths labelled may be empty.
The first two columns of a table indicate in which path labelled the edge or is located within the corresponding figure (symmetric cases have been omitted). The heading “” indicates the equation for a walk around the unique circuit contained in . The indicates the sign of the path labelled (with in the column with heading indicating that the path can be positive or negative), and is short for the product . A path can only be empty if its sign is positive. In the column “other cycles” the additional circuits (containing both and ) are given in terms of the equation (30) they define; the equality indicates how the equations for the circuit walks in and determine the equation given by each of these other circuit walks. Absence of a horizontal separator in a given column indicates that the assumption stated in the cell for the column is the same for the whole horizontal block.
Each expression is related to an oriented arc in the figure. If the arc is oriented from the vertex to the vertex then it is associated to a walk . The edges are then oriented following the orientation of the arc in such a way that
[TABLE]
(notice that all the signs of this expression are positive). In other words, the half-edge corresponding to attached to is always pointing from to . The remaining half-edges (which have a natural ordering from to ) point in a direction consistent with the previous half-edge:
- •
If is the second half-edge of an edge, then points in the same direction as the first half-edge if the edge is positive, and in the opposite direction if the edge is negative.
- •
If is the first half-edge of an edge, then it points in the same direction as the second half-edge of the previous edge (thus, at each internal vertex of the path the two incident half-edges are aligned).
Therefore, if are paths in which the terminal vertex of one path is the initial vertex of the next path, then the height of their concatenation is given by
[TABLE]
where
[TABLE]
Connection with Chen and Wang [9].
The coefficients in front of the expression in Tables 3-7 are integers and give, up to a sign, the coefficients of the indicator vectors of the circuits in [9] (denoted by in [9, (3.2)]). Then the equations given in (30) are those in [9, (4.4)]. By introducing cuts ([9, Section 2]) and bonds (following Zaslavsky’s [42, 41]), which are minimal cuts, they show that, as subspaces of , the cut space is orthogonal to the circuit space [9, Lemma 3.3], and giving a dimensional argument they show that the -vector subspace of generated by the indicator vectors of the circuits is of dimension [9, Theorem 3.5]. A basis is given by the indicator vectors of the circuit walks in , , where is a connected basis; the expression of any other circuit walk as a linear combination of the can be thus obtained using the coefficients of in the coordinates given by the edges in . This shows that the coefficients of the linear combination are integers (as, restricted to the coordinates given by , these indicator vectors of the give a diagonal matrix with as diagonal entries), and therefore the equations (30) for circuit walks can be deduced from those for the circuit walks alone. This gives a proof of Lemma 7.8; alternatively, the proof of Lemma 7.8 given earlier yields a proof of the result on the dimension of the circuit space from [9] that does not require the introduction of bonds.
Appendix C Dictionary of polynomials
Table 8 below gives details of the graph polynomials that feature or are mentioned in this paper, and Table 9 describes some relations between them.
Notation:
- •
: number of connected components
- •
: matroid rank; , rank of the set in matroid
- •
: vertex set of graph or signed graph
- •
: edge set of graph or signed graph, ground set of matroid
- •
, complement of in , or
- •
: spanning graph .
- •
/: number of unbalanced/balanced connected components of signed graph
- •
, and similarly , infinite vector of variables.
- •
: map obtained from by deleting edges not in (see [19] or [20, Definition A.1] for edge deletion in maps); the surface in which is embedded may differ from that of .
- •
: map obtained from by contracting edges in (see [20, Definition A.3] for edge contraction in maps – notably, contracting a loop may create an extra vertex); the surface in which is embedded may differ from that of ).
- •
: signed genus of connected component of given submap of , equal to the number of handles if orientably embedded, and minus the number of cross-caps if is non-orientably embedded.
- •
- •
where is the number of faces of the map , and is the number of vertices of the map .
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