While a there have been remarkable techniques and discoveries for representability over finite fields (see [GGW14; W05] for a selection), not much has been said about representability over infinite fields — except perhaps how badly our existing techniques fail. This blogpost will cover some of these failures and some hopeful conjectures (or future pathologies).
We use $\mathcal{M}(\mathbb{F})$ to denote the matroids representable over a field $\mathbb{F}$.
Excluded minors
For finite fields, we of course have the following result announced by Geelen, Gerards, and Whittle:
Theorem (Geelen, Gerards, Whittle [GGW14]):For each finite field $\mathbb{F}$, there are finitely many excluded minors for $\mathcal{M}(\mathbb{F})$.
This has many useful consequences. For each finite field $\mathbb{F}$, there is a constant time certificate when a matroid is not $\mathbb{F}$-representable and there is a finite second-order logical sentence for $\mathbb{F}$-representability using a predicate for independence.
In contrast, Mayhew, Newman, and Whittle showed that [MNW09]:
Theorem (Mayhew, Newman, Whittle [MNW09]): For each infinite field $\mathbb{F}$, each element of $\mathcal{M}(\mathbb{F})$ is a minor of an excluded minor for $\mathcal{M}(\mathbb{F})$.
In essence, the set of excluded minors for $\mathcal{M}(\mathbb{F})$ is as wild as $\mathcal{M}(\mathbb{F})$ itself. When the set of excluded minors for a class $\mathcal{C}$ contain $\mathcal{C}$ in its downwards-closure like this, we say that $\mathcal{C}$ is swamped. Another way in which the set of excluded minors for a class can be seen as “worse” then the class itself is when they dominate in quantity on a given number of elements. Taking $e_n$ to be the number of non-isomorphic excluded minors for $\mathcal{C}$ on at most $n$ elements and $c_n$ to be the number of non-isomorphic members of $\mathcal{C}$ on at most $n$ elements, we say that $\mathcal{C}$ is fractal when $\lim_{n\to\infty}\frac{e_n}{c_n}=\infty$. Mayhew, Newman, and Whittle show that for any integer $k\geq 3$, the class $\mathcal{C}$ of sparse paving matroids with at most $k$ circuit-hyperplanes is fractal [MNW2]. However, they leave the following open:
Problem: For each infinite field $\mathbb{F}$, is the class $\mathcal{M}(\mathbb{F})$ is fractal?
One of the most significant recent results in matroid representability theory is due to Nelson:
Theorem (Nelson [N18]):For $n\geq 12$, there are at most $2^{n^3/4}$ representable matroids on ground set $[n]$.
Combining this with Knuth’s upper bound ($2^{2^n/\text{poly}(n)}$) on the number of matroids on $[n]$, it gives us that asymptotically almost all matroids are non-representable. Along with this result, Nelson also gave one of the most significant conjectures in matroid representation theory:
Conjecture:Taking $d(n)=(\lfloor\frac{n}{2}\rfloor-1)(\lceil\frac{n}{2}\rceil-1)$, we have the following:
Asymptotically almost all representable matroids on $[n]$ have rank in $\{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil\}$ and has exactly $d(n)-1$ nonbases.
Asymptotically almost all matroids on $[n]$ with rank in $\{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil\}$ and exactly $d(n)-1$ nonbases are representable.
The value $d(n)-1$, is the maximum number of non-bases that can exist without enforcing a non-trivial algebraic relation and by only having points placed “generically”. The problem presented above is also interesting under the assumption that Nelson’s conjecture holds.
Natural Classes
Representability over any field is closed under minors and direct sums. However, a significant difference between finite and infinite fields is that with infinite fields we can place points in “general position”; matroid representability over an infinite field is closed under principal extension (i.e. freely placing an element into a flat). In general, we we say a class is natural, when it contains $U_{1,1}$ and is closed under the following: isomorphism, minors, direct sums, and principal extensions. It is the freedom of principal extension that seems to make natural classes wild. Besides, $\mathcal{M}(\mathbb{F})$ for $\mathbb{F}$ infinite, other natural classes include algebraic matroids over a given field, orientable matroids, and arbitrary intersections of natural classes. Also of particular note, is the class of Gammoids (matroids given by a linkage function in a directed graph), the smallest natural class. It is known that all of the above classes are swamped.
Problem:Are all proper natural classes swamped?
Problem:Are all proper natural classes fractal? Or are Gammoids swamped but not fragile?
Problem:Are asymptotically almost all representable matroids gammoids?
Efficient Discription
In fact it is open whether or not we have compact, efficient discription of matroids representable over a fixed infinite field. Considering the case of complex representable matroids:
Problem:Is there an algorithm $\mathcal{A}$ and polynomials $s$ and $t$, so that given a complex representable matroid $M=(E,r)$ there is a binary string description $\mathbf{D}_M$ (encoding $M$ in whatever which way), so that:
$\mathbf{D}_M$ has length at most $s(|E|)$, and
for any $X\subseteq E$, when $\mathcal{A}$ is given $(\mathcal{D}_M,X)$, it computes $r(X)$ in time at most $t(|E|)$.
For example, it would be sufficient to show that each complex representable matroid $M=(E,r)$ is representable by a matrix whose entries are algebraic with minimal polynomials whose degrees and coefficients are bounded by a polynomial in $|E|$. The best known bound on the degree is $2^{2^{2|E|^2}}$ given by Bell, Funk, Kim, and Mayhew [BFKW20]. Note that if $\mathbf{D}_M$ is a naive encoding of $r:2^E\to \mathbb{Z}$ (say as its list of outputs), then the running time $t$ is indeed polynomial, but the size $s$ is exponential. Alternatively, we can make the size $s$ polynomial with an enumeration of the complex representable matroids (but then $\mathcal{A}$ would certainly not run in polynomial time as it would have to reconstruct all prior matroids, and check each of these exponentially many rank evaluations). However, note that with a positive resolution to Nelson’s conjecture, we would be able to resolve this problem for almost all representable matroids by simply providing the size, rank, and nonbases.
I thank Geoff Whittle and Jim Geelen for helpful discussion on this post.
References
[BFKW20] B. Jason, F. Daryl, B. D. Kim, D. Mayhew, Effective Versions of Two Theorems of RADO, Quarterly J. Math. 71 (2020), 599–618.
[GGW14] J. Geelen, B. Gerards, G. Whittle, Solving Rota’s conjecture, Notices Amer. Math. Soc. 61 (2014), 736–743.
[MNW09] D. Mayhew, M. Newman, G. Whittle, On excluded minors for real representability, J. Combin. Theory Ser. B 99 (2009), 685–689.
[MNW21] D. Mayhew, M. Newman, G. Whittle, Fractal classes of matroids, Adv. in Appl. Math. 126 (2021), 101995.
[N18] P. Nelson, Almost all matroids are nonrepresentable, Bull. London Math. Soc. 50 (2018), 245-248.
[W05] G. Whittle, Recent work in matroid representation theory, Discrete Math. 302 (2005), 285–296.
This is a guest post by Donggyu Kim, who recently posted about the cycles of ribbon graphs. In this post, Donggyu introduces a generalisation of orientable ribbon graphs, pseudo-orientable ribbon graphs, based on recent work with Changxin Ding [2].
In my previous post, we saw that orientable ribbon graphs admit principally unimodular matrix representations. More precisely, for any orientable ribbon graph $\mathbb{G}$ and quasi-tree $X$, there exists a principally unimodular matrix $\mathbf{A}$ such that
$$\det(\mathbf{A}[Q \triangle X]) =\begin{cases}1 & \text{if $Q$ is a quasi-tree},\\0 & \text{otherwise},\end{cases}\qquad\qquad\tag{$*$}$$
where $Q\triangle X:= (Q\setminus X) \cup (X\setminus Q)$ denotes the symmetric difference of $Q$ and $X$, and $\mathbf{A}[Q \triangle X]$ is the principal submatrix of $\mathbf{A}$ indexed by $Q \triangle X$.
As noted in the previous post, the construction of such a matrix $\mathbf{A}$ relies heavily on the orientability of $\mathbb{G}$, and indeed it is not hard to find non-orientable ribbon graphs that do not admit such PU matrix representations.
Today, I will introduce a new class of ribbon graphs, called pseudo-orientable, which generalizes orientable ribbon graphs and admits PU matrix representations satisfying (*). For simplicity, we will restrict our attention to bouquets, that is, ribbon graphs with a single vertex. However, the results extend easily to general ribbon graphs by making use of the partial duality introduced by Chmutov [1]. We will also identify bouquets with signed chord diagrams; see Figure 1.
A bouquet (left) with three edges, where the edge $1$ is twisted/non-orientable (red) and the edges $2$ and $3$ are untwisted/orientable (blue). The corresponding signed chord diagram (right) is obtained by drawing chords on the boundary of the vertex and assigning signs (in this figure, the colors red and blue) to the chords according to the orientability of the corresponding edges.
Pseudo-orientability
Let’s begin by defining pseudo-orientable bouquets.
Definition. A bouquet $\mathbb{G}$ is pseudo-orientable if the boundary of the vertex admits two closed segments $S_1$ and $S_2$ such that
$S_1 \cap S_2$ is the set of two distinct points,
for each orientable edge, both of its ends lie in the interior of the same segment, either $S_1$ or $S_2$, and
for each non-orientable edge, one end lies in the interior of $S_1$ and the other end lies in the interior of $S_2$.
See Figure 2 (left) for an example of a pseudo-orientable bouquet.
The adjustment $\widehat{\mathbb{G}}$ of $\mathbb{G}$ at $(S_1,S_2)$ is the orientable bouquet obtained in the following way:
Cut the bouquet along $S_2$ and then reglue it with a half-twist. Note that all edges are orientable in the new bouquet.
Add a new orientable edge, denoted by $\widehat{e}$, connecting the two points in $S_1\cap S_2$.
Figure 2 (right) illustrates the resulting orientable bouquet $\widehat{\mathbb{G}}$.
The left figure is a pseudo-orientable bouquet $\mathbb{G}$, whose orientable edges are colored blue and non-orientable edges are colored red. A pair $(S_1,S_2)$ is indicated by dotted arrows. The right figure is the adjustment $\widehat{\mathbb{G}}$ of $\mathbb{G}$ at $(S_1,S_2)$, obtained by flipping the bottom segment $S_2$. The three blue-red dashed lines indicate orientable edges that were non-orientable in $\mathbb{G}$. The new edge $\widehat{e}$ is depicted as the blue dashed line. All edges in $\widehat{\mathbb{G}}$ are orientable.
Every orientable bouquet is pseudo-orientable, because we can get a pair $(S_1,S_2)$ by letting $S_1$ be a small boundary segment containing no ends of edges, and $S_2$ be the remaining boundary segment.
We also remark that the class of pseudo-orientable ribbon graphs is closed under taking ribbon graph minors.
PU matrix representations for pseudo-orientable bouquets
The Matrix–Quasi-tree Theorem for pseudo-orientable bouquets is as follows.
Theorem. Let $\mathbb{G}$ be a pseudo-orientable bouquet. Then there is an $E(\mathbb{G})$-by-$E(\mathbb{G})$ matrix $\mathbf{M}$ such that
$$\det(\mathbf{M}[Q])=\begin{cases}1 & \text{if $Q$ is a quasi-tree},\\0 & \text{otherwise}.\end{cases}$$
In particular, $\det(\mathbf{I} + \mathbf{M})$ is the number of quasi-trees of $\mathbb{G}$.
Remarkably, the matrix $\mathbf{M}$ is neither skew-symmetric nor symmetric in general, but can be expressed as the sum of a skew-symmetric matrix and a rank-one symmetric matrix. The theorem follows from two parallel stories on ribbon graphs and matrices, with delta-matroids underlying both.
For a subset $S$ of $[n]:=\{1,2,\ldots,n\}$, let
$$\widehat{S} :=\begin{cases}S & \text{if $|S|$ is even},\\S\cup\{n+1\} & \text{if $|S|$ is odd}.\end{cases}$$
Ribbon graphs. Let $\mathbb{G}$ be a pseudo-orientable bouquet, and let $\widehat{\mathbb{G}}$ be an adjustment of $\mathbb{G}$. We set $E(\mathbb{G}) = [n]$ and $\widehat{e} = n+1$. Then the following equivalence holds:
$$\text{$Q$ is a quasi-tree of $\mathbb{G}$} \iff \text{$\widehat{Q}$ is a quasi-tree of $\widehat{\mathbb{G}}$}.\tag{1}$$
Matrices. Let $\mathbf{A}$ be an $(n+1)$-by-$(n+1)$ PU skew-symmetric matrix representing $\widehat{\mathbb{G}}$. We can write it as
By the property (*) of the matrix $\mathbf{A}$ representing $\widehat{\mathbb{G}}$ (with $X=\emptyset$) together with (1) and (2), we deduce the theorem.
Delta-matroids. Delta-matroids are already lurking in the two pictures above. Recall that a ribbon graph gives rise to a delta-matroid via its quasi-trees, and a (skew-)symmetric matrix gives rise to a delta-matroid via the index sets of nonsingular principal submatrices. Moreover, the delta-matroid associated with an orientable ribbon graph is even. What, then, can we say about the delta-matroid associated with a pseudo-orientable ribbon graph?
A delta-matroid $D$ is strong if it satisfies the simultaneous basis exchange property: for any bases $B,B’$ and $x\in B\triangle B’$, there is $y\in B\triangle B’$ such that $B\triangle\{x,y\}$ and $B’\triangle\{x,y\}$ are bases.
Every matroid, even delta-matroid, and ribbon-graphic delta-matroid is strong, but not every delta-matroid is. For example, the delta-matroid $([3],\{\emptyset,\{1\},\{2\},\{3\},\{1,2,3\}\})$ is not strong.
Murota [4] gave a concise relationship between strong and even delta-matroids, attributing it to Geelen:
Proposition. Let $D = ([n],\mathcal{B})$ be a set system. Then $D$ is a strong delta-matroid if and only if $\widehat{D} := ([n+1],\widehat{\mathcal{B}})$ is an even delta-matroid, where $\widehat{\mathcal{B}} := \{ \widehat{B} : B\in\mathcal{B} \}$.
Finally, pseudo-orientable ribbon graphs are precisely those ribbon graphs $\mathbb{G}$, up to ribbon graph $2$-isomorphism [3], for which $\widehat{D(\mathbb{G})}$ is ribbon-graphic — this characterization captures our original motivation for the study [2].
Acknowledgements
I thank Changxin Ding and Jorn van der Pol for helpful comments.
References
[1] Sergei Chmutov. Generalized duality for graphs on surfaces and the signed Bollobas–Riordan polynomial. J. Combin. Theory Ser. B 99(3):617–638, 2009. doi:10.1016/j.jctb.2008.09.007 [2] Changxin Ding and Donggyu Kim. Pseudo-orientable ribbon graphs: Matrix–Quasi-tree theorem and log-concavity, 2026. arXiv preprint. [3] Iain Moffatt and Jeaseong Oh. A 2-isomorphism theorem for delta-matroids. Adv. in Appl. Math. 126, Paper No. 102133, 2021. doi:10.1016/j.aam.2020.102133 [4] Kazuo Murota. A note on M-convex functions on jump-systems. Discrete Appl. Math. 289:492–502, 2021. doi:10.1016/j.dam.2020.09.019
Many different areas of combinatorics and optimization are interested in objects that admit a rough duality between packing and covering. A classic example is that every bipartite graph either has a matching of size $k$, or has a set of at most $k-1$ vertices which “hit” every edge (i.e. a vertex cover of size $k$; this is Kőnig’s theorem).
Another example where only a rough duality is possible was proven by Erdős and Pósa. They showed that every graph either has $k$ cycles that are pairwise vertex-disjoint, or has a set of $\mathcal{O}(k \log{k})$ many vertices which hit every cycle (i.e. a set of vertices whose deletion yields a forest). Note that this is not an exclusive “or”; some graphs have both. But any graph with a hitting set of size $r$ has at most $r$ pairwise vertex-disjoint cycles.
In general, for any problem which is suitable to study within this framework, the maximum size of a packing should be at most the minimum size of a covering (or at least roughly so). This is typically the easy direction (I would love to hear in the comments about any natural examples where it is not the easy direction!). If the other direction is also (roughly) satisfied, then the set of objects is said to have the Erdős-Pósa property, meaning that if there is no packing of size $k$, then there is a covering of size at most $f(k)$, for some function $f$.
Related Topics
I have discussed examples of exact combinatorial min-max theorems before, and how these problems often have an underlying matroid. One of the most general theorems about which hypergraphs admit a rough duality between packing and covering is due to Ding, Seymour, and Winkler and generalizes well-known results on VC-dimension. Other key terms are LP-rounding, total unimodularity, clutters, … . Erdős-Pósa problems are very actively researched; see for instance the recent workshop in Poland. Geelen and Kabell also proved a different matroidal Erdős-Pósa property.
Binary Matroids
For binary matroids there is a somewhat different notion of covering. It is convenient to think of a graph with a “small vertex cover” as being “close to” a graph whose largest packing has size zero. So instead of deleting a small set of vertices, we add a low-rank matrix. Formally, consider a binary matroid $M$ which is represented by the columns of an $n \times m$ binary matrix $A$. A rank-$k$ perturbation of $M$ is then any matroid $M’$ which can be represented by the columns of $A+B$, where $B$ is an $n \times m$ binary matrix of rank at most $k$. We perform addition over the binary field.
Theorem For any planar graph $H$, there is a function $f_H$ so that for every integer $k$, every binary matroid either has the cycle matroid of $k$ vertex-disjoint copies of $H$ as a minor, or has a rank-$f_H(k)$ perturbation which does not have $M(H)$ as a minor.
This really is an Erdős-Pósa property in the sense that it gives a rough min-max theorem for packing $M(H)$ minors. When $H$ is a cycle and the matroid is graphic, one should roughly think of the theorem as saying the following. Note that this is a conjecture; it does NOT just follow from our theorem. This is because, informally, our perturbations are allowed to leave the graphic world, and the ones below are not.
Conjecture There exists a function $f$ so that for any integer $k$, every multigraph either 1) has a minor with $k$ different blocks, each of which is a cycle, or 2) can be turned into a forest by performing the following operations at most $f(k)$ times:
identify two vertices $u$ and $v$ into a single vertex. This operation does not change the number of edges. So for instance if $u$ and $v$ were adjacent originally, there will be a loop at the new vertex after identification.
split a vertex $v$ into two vertices $v_1$ and $v_2$. This operation is the opposite of identification.
Thank you to Mathieu Rundström for correcting the previous wrong version of the conjecture!
This is a guest post by Donggyu Kim, who discusses the cycles of an embedded graphs (ribbon graphs) and their interactions with the associated delta-matroids. It is based on joint work with Matt Baker and Changxin Ding [1].
Ribbon graphs and quasi-trees
An embedded graph $G$ is a graph cellularly embedded in a closed (possibly non-orientable) surface $\Sigma$. We think of vertices and edges as disks and rectangles, obtained by thickening the points and lines by a sufficiently small $\epsilon>0$ in the surface $\Sigma$. In this sense, embedded graphs are often called ribbon graphs; we will use the two notions interchangeably.
A typical example of an embedded graph is a connected plane graph – but instead of embedding the graph in the plane, let’s embed it on the sphere! One easy exercise is that a spanning subgraph $T$ is a spanning tree if and only if it, viewed as a ribbon graph, has a unique boundary component. However, these two notions diverge when the surface $\Sigma$ is not the sphere. Every spanning tree has a unique boundary component, but the converse doesn’t hold. We call a spanning subgraph with a unique boundary component a quasi-tree. The figure below depicts an embedded graph on the torus $\mathbb{T}^2$; it has a quasi-tree that is not a spanning tree. Since quasi-trees capture topological information, they are the right analog of spanning trees for embedded graphs.
Figure 1. A plane graph (an embedded graph on the sphere $\mathbb{S}^2$, left) and the corresponding ribbon graph (middle). The spanning subgraph with edge set $\{1,4,5\}$ (top right) has a unique boundary component as a ribbon graph, whereas the spanning subgraph with edge set $\{1,2,5\}$ (bottom right) has three boundary components – don’t miss the boundary of the degree-$0$ vertex.
Ribbon cycles
Question. We have seen that quasi-trees are the right analog of spanning trees for embedded graphs. What should play the role of cycles?
Before answering the question, let’s fix a convention. Since we’ll only care about the topological properties – such as whether $\Sigma – H$ is connected for a subgraph $H$ (we’ll make this more precise below) – every subgraph in this post will be assumed to be spanning (i.e., containing all vertices of the original graph). Hence, we will identify an edge set with the corresponding spanning subgraph. Moreover, we may view $\Sigma – H$ either as removing $H$ as a graph or as a ribbon graph. These two viewpoints are homeomorphic, so the reader may adopt whichever is more convenient.
With this convention in place, let’s return to the plane case (Figure 1). For any cycle $C$, the surface $\Sigma – C$ is disconnected; indeed, the cycles are exactly the minimal subgraphs with this property. What happens for general embedded graphs? Consider Figure 2. Even after removing any cycle from the torus $\mathbb{T}^2$, it remains connected. Moreover, there is no subgraph $H$ for which $\Sigma – H$ is disconnected.
Figure 2. A graph embedded on the torus (left) and the corresponding ribbon graph (right). It has four quasi-trees with edge sets $\{1\}$, $\{2\}$, $\{3\}$, and $\{1,2,3\}$.
Ordinary cycles alone cannot be the right answer on a general surface. To fix this, we need to bring the dual graph into the picture. Let us draw both the graph $G$ embedded on the torus, shown in Figure 2, and its geometric dual $G^*$ simultaneously; see Figure 3. The graph $G$ is drawn in black, and the dual $G^*$ is drawn in red. We can then separate the torus $\mathbb{T}^2$ into two components by removing a cycle $\{1,2\}$ of $G$ together with a cycle $\{3^*\}$ of $G^*$.
Figure 3. The embedded graph $G$ in Figure 2 together with its dual $G^*$. It has four ribbon cycles: $\{1,2,3^*\}$, $\{1,2^*,3\}$, $\{1^*,2,3\}$, and $\{1^*,2^*,3^*\}$.
Denote $E:=E(G)$ and $E^*:= E(G^*) = \{e^*:e\in E\}$. A subset $S \subseteq E\cup E^*$ is a subtransversal if it contains at most one of $e$ or $e^*$ for each edge $e\in E$.
Definition. Let $G$ be an embedded graph on $\Sigma$. A ribbon cycle of $G$ is a subtransversal $C \subseteq E\cup E^*$ that minimally separates the surface $\Sigma$, meaning that $\Sigma – C$ is disconnected while $\Sigma – C’$ is connected for every $C’\subsetneq C$.
If $G$ is a plane graph, then the ribbon cycles are exactly the cycles and the stars of bonds. For example, the ribbon cycles of the plane graph in Figure 1 (see also Figure 4) are
$\{1,2,5\}$, $\{3,4,5\}$, $\{1,2,3,4\}$, $\{1^*,2^*\}$, $\{3^*,4^*\}$, $\{1^*,3^*,5^*\}$, $\{1^*,4^*,5^*\}$, $\{2^*,3^*,5^*\}$, and $\{2^*,4^*,5^*\}$.
Figure 4. The plane graph in Figure 1 (black) and its geometric dual (red).
The embedded graph on the torus in Figure 3 has four ribbon cycles:
$\{1,2,3^*\}$, $\{1,2^*,3\}$, $\{1^*,2,3\}$, and $\{1^*,2^*,3^*\}$.
Each ribbon cycle $C$ is a union of cycles in $G$ and cycles in $G^*$. In other words, $C\cap E$ forms an Eulerian subgraph of $G$, and $C\cap E^*$ forms an Eulerian subgraph of $G^*$. The ribbon cycle $\{1^*,2^*,3^*\}$ consists of three cycles – $\{1^*\}$, $\{2^*\}$, and $\{3^*\}$ – which share a common vertex.
The following is an amusing exercise, a counterpart of the orthogonality of cycles and bonds in ordinary graphs. For an edge $e$ of an embedded graph $G$, we denote $(e^*)^* = e$, and write $C^* := \{e^* : e\in C\}$ for any subset $C\subseteq E\cup E^*$.
Exercise (Orthogonality). Let $C_1$ and $C_2$ be two ribbon cycles of $G$. Then $|C_1\cap C_2^*| \ne 1$.
Circuits of ribbon-graphic delta-matroids
Now that we have a topological candidate for the “cycles” of a ribbon graph, the next question is whether the delta-matroid associated with the ribbon graph captures the same objects. Carolyn Chun explained earlier that the quasi-trees of an embedded graph $G$ form the bases of a delta-matroid, denoted $D(G)$.
A graph gives rise to its graphic matroid in (at least) two familiar ways: by collecting maximal acyclic sets, we obtain the bases; by collecting cycles, we obtain the circuits. Thus one may ask whether the ribbon cycles of an embedded graph $G$ are the circuits of $D(G)$. However, there are two hurdles. First, the ground sets do not match, $E$ vs. $E\cup E^*$. Second, the bases of a delta-matroid are not equicardinal—which raises a problem, since bases cannot in general be recovered from circuits if the circuits are defined as minimal subsets not contained in any basis.
These issues can be easily resolved by introducing symmetric matroids [2] (also known as 2-matroids; see Irene Pivotto’s post), which are a homogenized version of delta-matroids.
Let $D = (E,\mathcal{B})$ be a finite set system. Then $D$ is a delta-matroid if and only if $\mathrm{lift}(D) := (E\cup E^*, \mathcal{B}’)$ is a symmetric matroid, where $\mathcal{B}’ := \{B\cup (E\setminus B)^* : B\in \mathcal{B}\}$. Since all bases of a symmetric matroid have the same cardinality, it makes sense to define its circuits. A circuit of a symmetric matroid is a minimal subtransversal that is not contained in any basis.
Now we can state the following:
Theorem. The circuits of $\mathrm{lift}(D(G))$ are exactly the ribbon cycles of an embedded graph $G$.
To prove this, let us first see equivalent characterizations of quasi-trees. Recall that a quasi-tree $Q$ is a spanning subgraph with a unique boundary component. Hence, $\Sigma-Q$ also has a unique boundary component, which implies that $\Sigma-Q$ is connected. The following folklore result says that an even stronger statement holds.
Proposition. For $Q\subseteq E(G)$, the following are equivalent:
$Q$ is a quasi-tree of $G$.
$(E\setminus Q)^*$ is a quasi-tree of the dual $G^*$.
$\Sigma – Q – (E\setminus Q)^*$ is connected.
We also note the following corollary orthogonality.
Exercise. For any subtransversal $S\subseteq E\cup E^*$ such that $\Sigma – S$ is connected, there is a transversal $T$ (that is, a subtransversal of size $|E|$) containing $S$ such that $\Sigma – T$ is still connected.
Hence, quasi-trees $Q$ can be identified with “maximal subtransversals $Q\cup (E\setminus Q)^*$ whose removal does not disconnect the surface,” whereas ribbon cycles are “minimal subtransversals whose removal disconnects the surface.” This proves the theorem above. Great! Ribbon cycles really are the right circuit-like objects.
Representations of ribbon-graphic delta-matroids via signed ribbon cycles
A natural next question is whether other familiar graph/matroid correspondences also survive in the ribbon-graph world. For example, if $G$ is an ordinary connected graph with an arbitrary orientation, we can quickly find a regular representation of $M(G)$ by examining the fundamental bonds with respect to a spanning tree:
Fix a spanning tree $T$.
For each $e\in E(T)$, let $C_e$ be the unique bond in the complement of $T-e$, called the fundamental bond of $e$ with respect to $T$.
Since a reference orientation of $G$ is given, we can identify $C_e$ with a $(0,\pm 1)$-vector (of course, this is determined up to sign, and we may choose either sign).
Construct a real $(|V|-1)$-by-$|E|$ matrix $\mathbf{A}$ by taking the vectors $C_e$ as its rows.
Then $\mathbf{A}$ is a totally unimodular matrix representing $M(G)$. In the same way, one can construct a totally unimodular matrix representing $M^*(G)$ by examining the fundamental cycles.
Remarkably, almost the same construction works for “orientable” ribbon graphs. Fix one of the two orientations of $\Sigma$. Here we choose the counterclockwise orientation with respect to normal vectors, the so-called right-hand rule. Fix a reference orientation for $G$. Then we can assign the corresponding dual reference orientation to $G^*$; see the figure below.
Figure 5. For each oriented edge $e$, we assign the dual orientation of $e^*$ as depicted on the left. The illustration on the right shows a reference orientation of $G$ in Figure 3 together with the corresponding dual reference orientation of $G^*$.
The point of the next four steps is to construct a matrix representing $\mathrm{lift}(D(G))$ from signed ribbon cycles.
Fix a quasi-tree $Q$. Denote $B := Q\cup (E\setminus Q)^*$.
For each $e \in E(G)$, let $C_e$ be the ribbon cycle in $B\triangle\{e,e^*\}$, which always exists and is unique!
We identify $C_e$ with a $(0,\pm 1)$-vector $\mathbf{v}_e$ in $\mathbb{R}^{E\cup E^*}$ defined as follows:
$\Sigma-C_e$ separates the surface into two components. Choose one of them, say $\Sigma’$.
$C_e$ is exactly the boundary of $\Sigma’$. Because $\Sigma’$ is oriented, we obtain a $(0,\pm 1)$-vector $\mathbf{v}_e$ supported on $C_e$ by setting \[\mathbf{v}_e(f) =\begin{cases}+1 & \text{if the orientation of $f\in C_e$ agrees with that of $\Sigma’$}, \\-1 & \text{if the orientation of $f\in C_e$ is opposite to that of $\Sigma’$}, \\0 & \text{if $f\notin C_e$}.\end{cases}\]
Construct a real $|E|$-by-$2|E|$ matrix $\mathbf{A}$ by taking the vectors $\mathbf{v}_e$ as its rows.
Proposition. Let $X$ be a transversal of $E\cup E^*$. Then $X \cap E$ is a quasi-tree of $G$ if and only if the $|E|$-by-$|E|$ submatrix $\mathbf{A}[(E\cup E^*) \setminus X]$ with columns $(E\cup E^*) \setminus X$ is nonsingular; in fact, its determinant is $\pm 1$.
Let’s see an example in the following figure.
Figure 6. Choose a quasi-tree $Q = \{1,2,3\}$. Then the ribbon cycles defined in Step 2 are $C_1 = \{1^*,2,3\}$ (left), $C_2 = \{1,2^*,3\}$ (middle), and $C_3 = \{1,2,3^*\}$ (right). The removal of each ribbon cycle $C_i$ from the torus $\mathbb{T}^2$ disconnects it into two components. In Step 3(a), we color $\Sigma’$ yellow.
The vectors $\mathbf{v}_i$ defined in Step 3(b) are
The entries in the first row are indexed by $1,2,3$ from left to right, and the entries in the second row are indexed by $1^*,2^*,3^*$ from left to right. Thus, the resulting matrix $\mathbf{A}$ in Step 4 is
where the columns are indexed by $1,2,3,1^*,2^*,3^*$ in this order. You can check that the matrix $\mathbf{A}$ satisfies the proposition.
After multiplying some rows by $-1$ and rearranging the columns, the matrix $\mathbf{A}$ in the proposition can be written as a matrix of the form $\left( \begin{array}{c|c} \mathbf{I}_n & \mathbf{S} \end{array} \right)$, where $\mathbf{I}_n$ is the identity matrix with size $n:=|E|$ and $\mathbf{S}$ is a principally unimodular skew-symmetric matrix. An equivalent construction of $\mathbf{S}$ was first given by Bouchet [2] in terms of (principally) unimodular orientations of circle graphs. If you are curious about the relationship between the above matrix construction and Bouchet’s construction, see Appendix A of [1]. Merino, Moffatt, and Noble [3] pointed out that
\[\det(\mathbf{I}_n + \mathbf{S}) = \# \text{quasi-trees of $G$},\]
The above matrix construction depends strongly on the orientability of the surface. This appears explicitly in Step 3, where we choose the signs of the entries of the $(0,\pm 1)$-vector $\mathbf{v}_e$. It is also implicit in Step 2, since we cannot guarantee the existence of $C_e$ when the surface is non-orientable.
Question. How about non-orientable ribbon graphs? Could we also obtain nice representations by unimodular matrices?
The short answer is: yes for some non-orientable ribbon graphs, but not for every ribbon graph. I will continue this discussion in the next post. See you then!
Acknowledgements. I thank Matt Baker, Changxin Ding, and Jorn van der Pol for helpful comments.
References
[1] Matthew Baker, Changxin Din, and Donggyu Kim. The Jacobian of a regular ortogonal matroid and torsor structures of quasi-trees of ribbon graphs, 2025. arXiv preprint.
[2] André Bouchet. Greedy algorithm and symmetric matroids. Math. Program. 38(2):147–159, 1987. doi:10.1007/BF02604639.
[3] Criel Merino, Iain Moffatt, and Steven Noble. The critical group of a combinatorial map. Comb. Theory 5(3), paper no. 2, 41, 2025. doi:10.5070/C65365550.
