Linear Algebraic Relations among Cardinalities of Sets of Matroid Functions

Abstract

We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs.

1. Introduction

A Tutte–Grothendieck invariant $\mathsf{\Phi }$ is a mapping from a class $\mathcal{M}$ of matroids to a commutative ring with the property that there are elements $a_1,b_1,a_2,b_2$ from the ring such that for a matroid $M\in \mathcal{M}$ on the ground set E, we have

$$\begin{array}{c}\hfill \begin{array}{ccc}& \mathsf{\Phi }\left(M\right)=1\hfill & \mathrm{if}E=\varnothing ,\hfill \\ & \mathsf{\Phi }\left(M\right)=a_1\mathsf{\Phi }(M-e)\hfill & \mathrm{if}e\mathrm{is}\mathrm{an}\mathrm{isthmus}\mathrm{of}M,\hfill \\ & \mathsf{\Phi }\left(M\right)=b_1\mathsf{\Phi }(M-e)\hfill & \mathrm{if}e\mathrm{is}\mathrm{a}\mathrm{loop}\mathrm{of}M,\hfill \\ & \mathsf{\Phi }\left(M\right)=a_2\mathsf{\Phi }(M/e)+b_2\mathsf{\Phi }(M-e)\hfill & \mathrm{otherwise}.\hfill \end{array}\end{array}$$

(Notice that $e\in E$ is an isthmus of M if it is contained in each base of M, and $e\in E$ is a loop of M if it belongs to no bases of M.) The best known Tutte–Grothendieck invariant is the Tutte polynomial:

$$\begin{array}{c}\hfill t(M;x,y)=\sum _{X\subseteq E}{(x-1)}^{r\left(M\right)-r_M\left(X\right)}{(y-1)}^{|X|-r_M\left(X\right)},\end{array}$$

which maps matroids to the ring of integral polynomials with two variables, x and y. This invariant was introduced by Tutte in [1] for graphs and encodes many properties of graphs and matroids. Applications of the Tutte polynomial in combinatorics, knot theory, statistical physics, and coding theory are surveyed in [2,3,4,5].

We consider classes of matroids whose ground sets contain a fixed subset B and study functions from the matroids to finite sets. For each of the matroids, consider the cardinality of a set of functions with fixed values on B. We show that if the cardinalities satisfy contraction–deletion rules, then there exist relations among these numbers that can be expressed in terms of linear algebra. In this way, we study numbers of regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs. These results generalize the approach that we introduced in [6,7].

2. Matroids and $\mathit{B}$-Classes

Throughout this paper, we consider finite matroids on finite sets. The ground set of a matroid M we denote by $E\left(M\right)$.

Let B be a finite set. A class of matroids $\mathcal{M}$ is called a B-class if

$$\begin{array}{c}\hfill \begin{array}{c}B\subseteq E\left(M\right)\mathrm{for}\mathrm{each}M\in \mathcal{M},\hfill \\ M-e,M/e\in \mathcal{M}\mathrm{for}\mathrm{each}M\in \mathcal{M}\mathrm{and}e\in E\left(M\right)\setminus B.\hfill \end{array}\end{array}$$

There exist only finitely many matroids on B. Thus, there exists a finite set ${\mathcal{M}}_B$ consisting of pairwise nonisomorphic matroids on B belonging to $\mathcal{M}$. (For example, if $\mathcal{M}$ is a class of matroids closed under contraction and deletion, then it is an ∅-class and ${\mathcal{M}}_{\varnothing }=\{\varnothing \}$.)

The collection of mappings from E to a finite set S is denoted by $S^E$. Assume that M is a matroid from $\mathcal{M}$ with the ground set $E=E\left(M\right)$. By an S-function on M, we mean any function $f\in S^E$. Then, $f|B$ denotes the restriction of f to B (i.e., $f|B\in S^B$ so that $[f|B\left]\right(x)=f(x)$ for each $x\in B$). Let $\mathcal{S}$ be a class of S-functions on matroids from $\mathcal{M}$. If $M\in \mathcal{M}$, then ${\mathcal{S}}_M$ denotes the set of S-functions on M belonging to $\mathcal{S}$. For every $g\in S^B$ and $M\in \mathcal{M}$, let

$$\begin{array}{c}\hfill \begin{array}{c}{\mathcal{S}}_{M,g}=\{f\in {\mathcal{S}}_M:f|B=g\}.\hfill \end{array}\end{array}$$

If $M\in {\mathcal{M}}_B$ and ${\mathcal{S}}_M=\varnothing$, then M is called $\mathcal{S}$-trivial; otherwise, it is called $\mathcal{S}$-nontrivial. In this paper, we denote by ${\mathcal{M}}_{B,\mathcal{S}}$ an ordered n-tuple $(M_1,\cdots ,M_n)$ of all $\mathcal{S}$-nontrivial elements of ${\mathcal{M}}_B$. For each $g\in S^B$ and $i=1,\cdots ,n$, let ${\chi }_{i,g}=1$ (${\chi }_{i,g}=0$) if $g\in {\mathcal{S}}_{B_i}$, ($g\notin {\mathcal{S}}_{B_i}$). Let ${\mathsf{\chi }}_{n,g}=({\chi }_{1,g},\cdots ,{\chi }_{n,g})$.

Let ${\mathbf{e}}_{i,n}$ denote the standard basis vector of ${\mathbb{R}}^n$ and let ${\mathbf{0}}_n$ denote the zero vector of ${\mathbb{R}}^n$. Vectors from ${\mathbb{R}}^n$ are considered row vectors. If $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$, then the dot product $\mathbf{x}·\mathbf{y}$ can be expressed as a matrix multiplication $\mathbf{x}{\mathbf{y}}^T$.

Assume that there exist rational numbers $a_1,b_1,a_2,b_2$ such that for each $M\in \mathcal{M}$, each $e\in E\left(M\right)\setminus B$, and each $g\in S^B$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{S}}_{M,g}|=a_1|{\mathcal{S}}_{M-e,g}|\hfill & \mathrm{if}e\mathrm{is}\mathrm{an}\mathrm{isthmus}\mathrm{of}M,\hfill \\ |{\mathcal{S}}_{M,g}|=b_1|{\mathcal{S}}_{M-e,g}|\hfill & \mathrm{if}e\mathrm{is}\mathrm{a}\mathrm{loop}\mathrm{of}M,\hfill \\ |{\mathcal{S}}_{M,g}|=a_2|{\mathcal{S}}_{M/e,g}|+b_2|{\mathcal{S}}_{M-e,g}|\hfill & \mathrm{otherwise}.\hfill \end{array}\end{array}$$

(1)

In this case, we say that $\mathcal{S}$ is $(a_1,b_1,a_2,b_2)$-regular. Since the cardinalities of sets are nonnegative integers, $a_1$, $b_1$ must be nonnegative, but only one of $a_2$, $b_2$ can be negative (but not both).

Theorem 1.

Let $\mathcal{M}$ be a B-class of matroids, with B finite, $\mathcal{S}$ be an $(a_1,b_1,a_2,b_2)$-regular class of S-functions on matroids from $\mathcal{M}$, with S finite, and ${\mathcal{M}}_{B,\mathcal{S}}=(M_1,\cdots ,M_n)$. Then, for each $M\in \mathcal{M}$, there exists a vector ${\mathbf{x}}_M=(x_1,\cdots ,x_n)$ such that for every $g\in S^B$, $|{\mathcal{S}}_{M,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_M$, i.e., $|{\mathcal{S}}_{M,g}|={\sum }_{i=1}^n{\chi }_{i,g}{x}_i$. Furthermore, if $a_1,b_1,a_2,b_2$ are integers, then ${\mathbf{x}}_M$ can be chosen to be an integral vector.

Proof. 

We apply induction on $|E(M)|\ge |B|$. Let $|E(M)|=|B|$ and $g\in S^B$. If M is $\mathcal{S}$-trivial, then $|{\mathcal{S}}_{M,g}|=0$, and we can set ${\mathit{x}}_M={\mathbf{0}}_n$. If M is $\mathcal{S}$-nontrivial, then $M=M_i$, where $i\in \{1,\cdots ,n\}$, and $|{\mathcal{S}}_{M,g}|={\chi }_{i,g}$. Thus, ${\mathit{x}}_M={\mathbf{e}}_{i,n}$ satisfies the assumptions.

If $|E(G)|>|B|$, then there exists $e\in E\left(M\right)\setminus B$. By the induction hypothesis, there are vectors ${\mathbf{x}}_{M-e}$ and ${\mathbf{x}}_{M/e}$ such that for every $g\in S^B$, $|{\mathcal{S}}_{M-e,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M-e}$ and $|{\mathcal{S}}_{M/e,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M/e}$.

If e is an isthmus, then from the first row of (1), $|{\mathcal{S}}_{M,g}|=a_1|{\mathcal{S}}_{M-e,g}|=a_1{\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M-e}$, where the vector ${\mathbf{x}}_M=a_1{\mathbf{x}}_{M-e}$ satisfies the assumptions.

If e is a loop, then from the second row of (1), $|{\mathcal{S}}_{M,g}|=b_1|{\mathcal{S}}_{M-e,g}|=b_1{\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M-e}$, where the vector ${\mathbf{x}}_M=b_1{\mathbf{x}}_{M-e}$ satisfies the assumptions.

If e is neither an isthmus nor a loop, then from (1), $|{\mathcal{S}}_{M,g}|=a_2|{\mathcal{S}}_{M/e,g}|+b_2|{\mathcal{S}}_{M-e,g}|=a_2{\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M/e}+b_2{\mathsf{\chi }}_{n,g}·{\mathbf{x}}_{M-e}={\mathsf{\chi }}_{n,g}·(a_2{\mathbf{x}}_{M/e}+b_2{\mathbf{x}}_{M-e})$, where the vector ${\mathbf{x}}_M=a_2{\mathbf{x}}_{M/e}+b_2{\mathbf{x}}_{M-e}$ satisfies the assumptions.

If $a_1,b_1,a_2,b_2$ are integers, then all vectors ${\mathbf{x}}_M$ considered in the proof are integral. This proves the statement. □

Let $Z=({\mathbf{z}}_1,\cdots ,{\mathbf{z}}_m)$, $m\le n$, be an ordered basis of the linear hull of $\{{\mathsf{\chi }}_{n,g};g\in S^B\}$. Denote ${\mathsf{\chi }}_{Z,g}=(t_1,\cdots t_m)$ such that ${\mathsf{\chi }}_{n,g}={\sum }_{i-1}^m{t}_i{\mathbf{z}}_i$. For example, if $n=m$, then we can choose ${\mathbf{z}}_i={\mathbf{e}}_{i,n}$ and then ${\mathsf{\chi }}_{Z,g}={\mathsf{\chi }}_{n,g}$.

Theorem 2.

Let $\mathcal{M}$ be a B-class of matroids, with B finite, $\mathcal{S}$ be an $(a_1,b_1,a_2,b_2)$-regular class of S-functions on matroids from $\mathcal{M}$, with S finite, ${\mathcal{M}}_{B,\mathcal{S}}=(M_1,\cdots ,M_n)$, and $Z=({\mathbf{z}}_1,\cdots ,{\mathbf{z}}_m)$, $m\le n$, be an ordered basis of the linear hull of $\{{\mathsf{\chi }}_{n,g};g\in S^B\}$. Then, for each $M\in \mathcal{M}$, there exists a unique vector ${\mathbf{y}}_M=(y_1,\cdots ,y_m)$ such that for every $g\in S^B$, $|{\mathcal{S}}_{M,g}|={\mathsf{\chi }}_{Z,g}·{\mathbf{y}}_M$. Furthermore:

• If $E\left(M\right)=B$ and M is trivial, then ${\mathbf{y}}_M={\mathbf{0}}_m$;
• If $E\left(M\right)=B$ and $M=M_i$, $i\in \{1,\cdots ,n\}$, then ${\mathbf{y}}_M=(y_1,\cdots ,y_m)$ such that $y_j$ is the i-th coordinate of ${\mathbf{z}}_j$, $j\in \{1,\cdots ,m\}$;
• If $E\left(M\right)\ne B$, then ${\mathbf{y}}_M$ satisfies the following recursive rules:

  $$\begin{array}{c}\hfill \begin{array}{cc}{\mathbf{y}}_M=a_1{\mathbf{y}}_{M-e}\hfill & \mathrm{if}e\mathrm{is}\mathrm{an}\mathrm{isthmus}\mathrm{of}M,\hfill \\ {\mathbf{y}}_M=b_1{\mathbf{y}}_{M-e}\hfill & \mathrm{if}e\mathrm{is}\mathrm{a}\mathrm{loop}\mathrm{of}M,\hfill \\ {\mathbf{y}}_M=a_2{\mathbf{y}}_{M/e}+b_2{\mathbf{y}}_{M-e}\hfill & \mathrm{otherwise}.\hfill \end{array}\end{array}$$

  (2)

Finally, if $a_1,b_1,a_2,b_2$ are integers and ${\mathbf{z}}_1,\cdots ,{\mathbf{z}}_m$ are integral vectors, then ${\mathbf{y}}_M$ is an integral vector for each M from $\mathcal{M}$.

Proof. 

We prove the existence of ${\mathbf{y}}_M$ by induction on $|E(M)|\ge |B|$. Let $|E(M)|=|B|$ and $g\in S^B$. If M is $\mathcal{S}$-trivial, then $|{\mathcal{S}}_{M,g}|=0$, and we can set ${\mathbf{y}}_M={\mathbf{0}}_m$. If M is $\mathcal{S}$-nontrivial and $M=M_i$, $i\in \{1,\cdots ,n\}$, then from the proof of Theorem 1, $|{\mathcal{S}}_{M,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{e}}_{i,n}$. Let A be an $m\times n$-matrix with the j-th row equal to ${\mathbf{z}}_j$, $j\in \{1,\cdots ,m\}$. Using matrix multiplication, we can express ${\mathsf{\chi }}_{n,g}={\mathsf{\chi }}_{Z,g}A$ and $|{\mathcal{S}}_{M,g}|={\mathsf{\chi }}_{n,g}{\mathbf{e}}_{i,n}^T$, where $|{\mathcal{S}}_{M,g}|=\left({\mathsf{\chi }}_{Z,g}A\right){\mathbf{e}}_{i,n}^T={\mathsf{\chi }}_{Z,g}\left(A{\mathbf{e}}_{i,n}^T\right)$. Thus, $A{\mathbf{e}}_{i,n}^T={\mathbf{y}}_M^T$, where ${\mathbf{y}}_M=(y_1,\cdots ,y_m)$ such that $y_j$ is the i-th coordinate of ${\mathbf{z}}_j$, $j\in \{1,\cdots ,m\}$.

If $|E(G)|>|B|$, then there exists $e\in E\left(M\right)\setminus B$. By the induction hypothesis, there are integral vectors ${\mathbf{y}}_{M-e}$ and ${\mathbf{y}}_{M/e}$ such that for every $g\in S^B$, $|{\mathcal{S}}_{M-e,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M-e}$ and $|{\mathcal{S}}_{M/e,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M/e}$.

If e is an isthmus, then from the first row of (1), $|{\mathcal{S}}_{M,g}|=a_1|{\mathcal{S}}_{M-e,g}|=a_1{\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M-e}$, where ${\mathbf{y}}_M=a_1{\mathbf{y}}_{M-e}$.

If e is a loop, then from the second row of (1), $|{\mathcal{S}}_{M,g}|=b_1|{\mathcal{S}}_{M-e,g}|=b_1{\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M-e}$, where ${\mathbf{y}}_M=b_1{\mathbf{y}}_{M-e}$.

If e is neither an isthmus nor a loop, then from (1), $|{\mathcal{S}}_{M,g}|=a_2|{\mathcal{S}}_{M/e,g}|+b_2|{\mathcal{S}}_{M-e,g}|=a_2{\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M/e}+b_2{\mathsf{\chi }}_{n,g}·{\mathbf{y}}_{M-e}={\mathsf{\chi }}_{n,g}·(a_2{\mathbf{y}}_{M/e}+b_2{\mathbf{y}}_{M-e})$, where the vector ${\mathbf{y}}_M=a_2{\mathbf{y}}_{M/e}+b_2{\mathbf{y}}_{M-e}$ satisfies the assumptions. This proves (2).

The uniqueness of ${\mathbf{y}}_M$ follows from the fact that Z is a basis of the linear hull of $\{{\mathsf{\chi }}_{n,g};g\in S^B\}$.

Furthermore, if ${\mathbf{z}}_1,\cdots ,{\mathbf{z}}_m$ are integral vectors, then ${\mathbf{y}}_M$ is integral for each $M\in \mathcal{M}$ such that $E\left(M\right)=B$. If $a_1,b_1,a_2,b_2$ are also integers, then from (2), ${\mathbf{y}}_M$ are integral vectors for each M from $\mathcal{M}$. □

We apply Theorem 1 for various S-functions of B-classes of matroids. Analogously, we can apply Theorem 2.

3. Regular Chain Groups

If R is a ring, the elements of $R^E$ are considered vectors indexed by E, and we will use the notation $f+g$, $-f$, and $sf$ for $f,g\in R^E$, and $s\in R$. A chain on E (over R, or simply an R-chain) is $f\in R^E$, and the support of f is $\sigma \left(f\right)=\{e\in E;f(e)\ne 0\}$. The zero chain has null support. Given $X\subseteq E$ and $f\in R^E$, define $f^{\setminus X}\in R^{E\setminus X}$ such that $\left[f^{\setminus X}\right]\left(e\right)=f\left(e\right)$ for each $e\in E\setminus X$.

A matroid M on E of rank $r\left(M\right)$ is regular if there exists an $r\times n$ ($r=r\left(M\right)$, $n=|E|$) totally unimodular matrix D (called a representative matrix of M) such that independent sets of M correspond to independent sets of columns of D.

We recall properties of regular matroids presented in [1,8,9,10,11]). For any basis B of M, D can be transformed to a form $(I_r|U)$ such that $I_r$ corresponds to B and U is totally unimodular. The dual of M is a regular matroid $M^{*}$ with a representative matrix $(-U^T|I_{n-r})$ (where $I_{n-r}$ corresponds to $E\setminus B$).

By a regular chain group N on E (associated with D), we mean a set of chains on E over $\mathbb{Z}$ that are orthogonal to each row of D (i.e., are integral combinations of rows of a representative matrix of $M^{*}$). The set of chains orthogonal to every chain of N is a chain group called orthogonal to N and denoted by $N^{\perp }$ (clearly, $N^{\perp }$ is the set of integral combinations of rows of D). By the rank of N, we mean $r\left(N\right)=n-r\left(M\right)=r^{*}\left(M\right)$. Then, $r\left(N^{\perp }\right)=n-r\left(N\right)=r\left(M\right)$. We always assume that a regular chain group N is associated with a matrix $D=D\left(N\right)$ representing a matroid $M=M\left(N\right)$.

For any $X\subseteq E$, let

$$\begin{array}{c}\hfill \begin{array}{c}N-X=\left\{f^{\setminus X};f\in N,\sigma \left(f\right)\cap X=\varnothing \right\},\hfill \\ N/X=\left\{f^{\setminus X};f\in N\right\}.\hfill \end{array}\end{array}$$

(3)

We have $M(N-X)=M-X$ and $M(N/X)=M/X$. Clearly, $D(N-X)$ arises from $D\left(N\right)$ after deleting the columns corresponding to X. Furthermore, ${(N-X)}^{\perp }=N^{\perp }/X$ and ${(N/X)}^{\perp }=N^{\perp }-X$.

A chain f of N is elementary if there is no nonzero $f^{\prime }$ of N such that $\sigma \left(f^{\prime }\right)\subset \sigma \left(f\right)$. An elementary chain f is called a primitive chain of N if the coefficients of f are restricted to the values 0, 1, and $-1$. (Notice that the set of supports of primitive chains of N is the set of circuits of $M\left(N\right)$.) We say that a chain gconforms to a chain f if $g\left(e\right)$ and $f\left(e\right)$ are nonzero and have the same sign for each $e\in E$ such that $g\left(e\right)\ne 0$. From [1] (Section 6.1),

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{every}\mathrm{chain}f\mathrm{of}N\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{a}\mathrm{sum}\mathrm{of}\\ \mathrm{primitive}\mathrm{chains}\mathrm{in}N\mathrm{that}\mathrm{conform}\mathrm{to}f.\end{array}\end{array}$$

(4)

Let A be an Abelian group with additive notation. We shall consider A as a (right) $\mathbb{Z}$-module such that the scalar multiplication $a·z$ of $a\in A$ by $z\in \mathbb{Z}$ is equal to 0 if $z=0$, ${\sum }_1^{z}a$ if $z>0$, and ${\sum }_1^{-z}(-a)$ if $z<0$. Similarly, if $a\in A$ and $f\in {\mathbb{Z}}^E$, then define $a·f\in A^E$ so that $(a·f)\left(e\right)=a·f\left(e\right)$ for each $e\in E$. If N is a regular chain group on E, define

$$\begin{array}{c}\hfill \begin{array}{c}A\left(N\right)=\left\{\sum _{i=1}^m{a}_i·f_i;a_i\in A,f_i\in N,m\ge 1\right\},\hfill \\ A\left[N\right]=\left\{f\in A\left(N\right);\sigma \left(f\right)=E\right\}.\hfill \end{array}\end{array}$$

Notice that $A\left(N\right)=N$ if $A=\mathbb{Z}$. From [8] (Proposition 1),

$$\begin{array}{c}\hfill \begin{array}{c}g\in A^E\mathrm{is}\mathrm{from}A\left(N\right)\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\mathrm{for}\mathrm{each}f\in N^{\perp },\\ \sum _{e\in E}g\left(e\right)f\left(e\right)=0.\end{array}\end{array}$$

(5)

Suppose that $\mathcal{M}$ is a B-class of regular matroids, with B finite. Denote by $\mathcal{R}$ the class of $A-\left\{0\right\}$-functions on matroids from $\mathcal{M}$ such that $A\left[N\right]={\mathcal{R}}_{M\left(N\right)}$ for each $M\left(N\right)\in \mathcal{M}$. In other words, $\mathcal{R}$ is the class of $A\left[N\right]$ where $M\left(N\right)\in \mathcal{M}$. We claim that $\mathcal{R}$ is $(0,k-1,1,-1)$-regular.

Lemma 1.

For each $M\in \mathcal{M}$, $e\in E\left(M\right)\setminus B$, and $g:B\to A-\left\{0\right\}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{R}}_{M,g}|=0|{\mathcal{R}}_{M-e,g}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}M,\hfill \\ |{\mathcal{R}}_{M,g}|=(k-1)|{\mathcal{R}}_{M-e,g}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}M,\hfill \\ |{\mathcal{R}}_{M,g}|=|{\mathcal{R}}_{M/e,g}|-|{\mathcal{R}}_{M-e,g}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

(6)

Proof. 

Notice that e is a loop (isthmus) of $M=M\left(N\right)$e if ${\chi }_e\in N$ (${\chi }_e\in N^{\perp }$). Thus, if e is an isthmus of M, then from (5), each $f\in {\mathcal{R}}_{M,g}$ satisfies $f\left(e\right)=0$, where $|{\mathcal{R}}_{M,g}|=0$.

Given $f\in A^{E\setminus e}$ and $x\in A$, let $f_x\in A^E$ be defined so that $f_x^{\setminus e}=f$ and $f_x\left(e\right)=x$.

If e is a loop of M, then from (5), for each $f\in {\mathcal{R}}_{M-e,g}$ and $x\in A-\left\{0\right\}$, $f_x\in {\mathcal{R}}_{M,g}$. Similarly, if $f\in {\mathcal{R}}_{M,g}$, then $f^{\setminus e}\in {\mathcal{R}}_{M-e,g}$. Thus, $|{\mathcal{R}}_{M,g}|=(k-1)|{\mathcal{R}}_{M-e,g}|$.

If e is neither an isthmus nor a loop of $M=M\left(N\right)$, then there exists $\tilde{f}\in N^{\perp }$ such that $\tilde{f}\left(e\right)\ne 0$ and $\tilde{f}\ne {\chi }_e$. From (3), for any $f\in A[N/e]$, there exists $a\in A$ such that $f_a\in A\left(N\right)$. From (5), $f_a$ must be orthogonal to $\tilde{f}$, where a is unique. Furthermore, if $a=0$ (resp. $a\ne 0$), then from (3), $f\in A[N-e]$ (resp. $f_a\in A\left[N\right]$); i.e., $f\mapsto f_a$ is a bijection from ${\mathcal{R}}_{M/e,g}$ to the disjoint union of ${\mathcal{R}}_{M,g}$ and ${\mathcal{R}}_{M-e,g}$. This implies the last row of (6). □

Corollary 1.

Suppose that $\mathcal{M}$ is a B-class of regular matroids, with B finite, and let $\mathcal{R}$ be the class of $A\left[N\right]$ where $M\left(N\right)\in \mathcal{M}$. Assume that ${\mathcal{M}}_{B,\mathcal{R}}=(M_1,\cdots ,M_n)$. Then, for each $M\in \mathcal{M}$, there exists an integral vector ${\mathbf{x}}_M=(x_1,\cdots ,x_n)$ such that for every $g:B\to A-\left\{0\right\}$, $|{\mathcal{R}}_{M,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_M$.

Proof. 

It follows from (6) and Theorem 1. □

4. Nowhere-Zero Flows and Tensions on Graphs

We deal with finite undirected graphs with multiple edges and loops. If G is a graph, then $V\left(G\right)$ and $E\left(G\right)$ denote its vertex and edge sets, respectively. Every edge e of G determines two opposite arcs arising from it after endowing e with two distinct orientations. All arcs obtained in this way are called arcs of G, and the set of them is called the arc set of G and denoted by $D\left(G\right)$. Clearly, $|D(G)|=2|E(G)|$. If x is an arc of G, then denote by $x^{-1}$ the second arc arising from the same edge. Clearly, ${\left(x^{-1}\right)}^{-1}=x$ and $x\ne x^{-1}$ for every arc x of G. If $X\subseteq D\left(G\right)$, then let $X^{-1}$ denote $\{x\in D\left(G\right);x^{-1}\in X\}$. For any vertex v of G, denote by ${\omega }_G^{+}\left(v\right)$ the set of arcs from $D\left(G\right)$ directed out of v. If A is an Abelian group, then a nowhere-zero A-chain in G is a mapping $\phi :D\left(G\right)\to A-\left\{0\right\}$ such that $\phi \left(x^{-1}\right)=-\phi \left(x\right)$ for every $x\in D\left(G\right)$.

By an orientation of G, we mean any $X\subseteq D\left(G\right)$ such that $X\cup X^{-1}=D\left(G\right)$ and $X\cap X^{-1}=\varnothing$. In other words, an orientation of G can be considered a directed graph arising from G after endowing each edge with an orientation.

Let A be an Abelian group with additive notation. A nowhere-zero A-chain $\phi$ in G is called a nowhere-zero A-flow if ${\sum }_{x\in {\omega }_G^{+}\left(v\right)}\phi \left(x\right)=0$ for every vertex v of G. Considering $\phi$ as a mapping on an arbitrary but fixed orientation of G, we obtain the usual definition of nowhere-zero A-flows. Such nowhere-zero A-flows on G coincide with $A\left[N\right]$, where N is the regular chain group associated with $M\left(N\right)$, the cycle matroid of G (edge sets of subforests of G forming independent sets of $M\left(N\right)$).

By a B-class of graphs, we mean a class $\mathcal{G}$ such that for each $G\in \mathcal{G}$, $B\subseteq E\left(G\right)$, and for each $e\in E\left(G\right)\setminus B$, $G-e,G/e\in \mathcal{G}$. Then, the class of cycle matroids of graphs from $\mathcal{G}$ is a B-class of matroids $\mathcal{M}$. Denote by $\mathcal{F}$ the class of nowhere-zero A-flows on a graph from $\mathcal{G}$. Clearly, $\mathcal{F}$ coincides with the class $\mathcal{R}$ associated with $\mathcal{M}$ described in the previous section. Analogously, we write ${\mathcal{G}}_{B,\mathcal{F}}=(G_1,\cdots ,G_n)$ instead of ${\mathcal{M}}_{B,\mathcal{R}}=(M_1,\cdots ,M_n)$, where $M_i$ is the cycle matroid of $G_i$ for $i=1,\cdots ,n$.

Lemma 2.

For each $G\in \mathcal{G}$, $e\in E\left(G\right)\setminus B$, and $g:B\to A-\left\{0\right\}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{F}}_{G,g}|=0|{\mathcal{F}}_{G-e,g}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}M,\hfill \\ |{\mathcal{F}}_{G,g}|=(k-1)|{\mathcal{F}}_{G-e,g}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}M,\hfill \\ |{\mathcal{F}}_{G,g}|=|{\mathcal{F}}_{G/e,g}|-|{\mathcal{F}}_{G-e,g}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

Proof. 

Apply Lemma 1 for a class $\mathcal{M}$ of cycle matroids of graphs from $\mathcal{G}$. □

Corollary 2.

Suppose that $\mathcal{G}$ is a B-class of graphs, with B finite, and let $\mathcal{F}$ be the class of nowhere-zero A-flows on graphs from $\mathcal{G}$. Assume that ${\mathcal{G}}_{B,\mathcal{F}}=(G_1,\cdots ,G_n)$. Then, for each $G\in \mathcal{G}$, there exists an integral vector ${\mathbf{x}}_G=(x_1,\cdots ,x_n)$ such that for every $g:B\to A-\left\{0\right\}$, $|{\mathcal{F}}_{G,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_G$.

Proof. 

Apply Corollary 1 for a class $\mathcal{M}$ of cycle matroids of graphs from $\mathcal{G}$. □

We applied the idea of Corollary 2 in [6,7,12,13] and proved that the smallest counterexample to the 5-flow conjecture of Tutte (that every bridgeless graph has a nowhere-zero 5-flow) must be cyclically 6-edge-connected and has a girth of at least 11.

A circuit C of G is a connected 2-regular subgraph of G (notice that the loop is a circuit of order 1). By a directed circuit of G, we mean an orientation X of C such that $|X\cap {\omega }_G^{+}\left(v\right)|=1$ for each vertex v of C.

A nowhere-zero A-chain $\phi$ in G is called a nowhere-zero A-tension if ${\sum }_{x\in X}\phi \left(x\right)=0$ for every directed circuit X of G. Considering $\phi$ as a mapping on an arbitrary but fixed orientation of G, we obtain nowhere-zero A-tensions on G that coincide with $A\left[N\right]$ such that $M\left(N\right)$ is the bond matroid of G (dual of the cycle matroid of G). Denote by $\mathcal{T}$ the class of nowhere-zero A-tensions on graphs from $\mathcal{G}$. Clearly, $\mathcal{T}$ coincides with the class $\mathcal{R}$ associated with the class of bond matroids of graphs from $\mathcal{G}$. Therefore, $\mathcal{T}$ is $(k-1,0,-1,1)$-regular.

Lemma 3.

For each $G\in \mathcal{G}$, $e\in E\left(G\right)\setminus B$, and $g:B\to A-\left\{0\right\}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{T}}_{G,g}|=(k-1)|{\mathcal{T}}_{G-e,g}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}M,\hfill \\ |{\mathcal{T}}_{G,g}|=0|{\mathcal{T}}_{G-e,g}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}M,\hfill \\ |{\mathcal{T}}_{G,g}|=|{\mathcal{T}}_{G-e,g}|-|{\mathcal{T}}_{G/e,g}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

Proof. 

Apply Lemma 1 for the class of bond matroids of graphs from $\mathcal{G}$. □

Corollary 3.

Suppose that $\mathcal{G}$ is a B-class of graphs, with B finite, and let $\mathcal{T}$ be the class of nowhere-zero A-tensions on graphs from $\mathcal{G}$. Assume that ${\mathcal{G}}_{B,\mathcal{T}}=(G_1,\cdots ,G_n)$. Then, for each $G\in \mathcal{G}$, there exists an integral vector ${\mathbf{x}}_G=(x_1,\cdots ,x_n)$ such that for every $g:B\to A-\left\{0\right\}$, $|{\mathcal{T}}_{G,g}|={\mathsf{\chi }}_{n,g}·{\mathbf{x}}_G$.

Proof. 

Apply Corollary 1 for the class of bond matroids of graphs from $\mathcal{G}$. □

5. Orientations in Oriented Matroids

In this section, we use notation and results from [14,15] (see also [9,16,17]). We define a signed set X to be a set $\underline{X}$, called the set underlying X, and the mapping ${\mathrm{sg}}_X\left(x\right):\underline{X}\to \{1,-1\}$, called the signature of X. Let X be a signed set. Then, $\underline{X}$ is partitioned into two distinguished subsets: $X^{+}=\{x\in \underline{X};{\mathrm{sg}}_X\left(x\right)=1\}$ and $X^{-}=\{x\in \underline{X};{\mathrm{sg}}_X\left(x\right)=-1\}$. The opposite$-X$ of X is defined by ${(-X)}^{+}=X^{-}$ and ${(-X)}^{-}=X^{+}$. If $\underline{X}$ is a subset of E, then X will be called a signed subset of E, and if $\underline{X}=\varnothing$, then we write $X=\varnothing$.

An oriented matroid M on E is a couple $(E,\mathcal{O})$, where $\mathcal{O}$ is a collection of signed sets satisfying

$$\begin{array}{c}\hfill X\in \mathcal{O}\mathrm{implies}X\ne \varnothing \mathrm{and}-X\in \mathcal{O};\end{array}$$

(7)

$$\begin{array}{c}\hfill X_1,X_2\in \mathcal{O}\mathrm{and}{\underline{X}}_1\subseteq {\underline{X}}_2\mathrm{imply}{X}_1=X_2\mathrm{or}{X}_1=-X_2;\end{array}$$

(8)

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{for}\mathrm{all}{X}_1,X_2\in \mathcal{O},x\in X_1^{+}\cap X_2^{-}\mathrm{and}y\in X_1^{+}\setminus X_2^{-}\mathrm{there}\mathrm{exists}{X}_3\in \mathcal{O}\hfill \\ \mathrm{such}\mathrm{that}y\in X_3,X_3^{+}\subseteq (X_1^{+}\cup X_2^{+})\setminus \left\{x\right\}\mathrm{and}{X}_3^{-}\subseteq (X_1^{-}\cup X_2^{-})\setminus \left\{x\right\}.\hfill \end{array}\end{array}$$

(9)

Signet sets from $\mathcal{O}$ are called signed circuits of M. Let $\underline{\mathcal{O}}=\{\underline{X};X\in \mathcal{O}\}$. Then, $\underline{\mathcal{O}}$ is a collection of circuits of a matroid $\underline{M}$ on E. The circuits of the dual matroid ${\underline{M}}^{*}$ (i.e., the cocircuits of $\underline{M}$) can be oriented in a unique way such that the ${\mathcal{O}}^{*}$ of signed cocircuits of M satisfies the orthogonality property: for all $X\in \mathcal{O}$ and $Y\in {\mathcal{O}}^{*}$ such that $|\underline{X}\cap \underline{Y}|=2$, both $(X^{+}\cap Y^{+})\cup (X^{-}\cap Y^{-})$ and $(X^{+}\cap Y^{-})\cup (X^{-}\cap Y^{+})$ are non-empty. Then, ${\mathcal{O}}^{*}$ satisfies (7)–(9) and defines an oriented matroid $M^{*}$, the dual of M. The orthogonality property holds for all $X\in \mathcal{O}$ and $Y\in {\mathcal{O}}^{*}$ such that $\underline{X}\cap \underline{Y}\ne \varnothing$. We have ${\left(M^{*}\right)}^{*}=M$. Thus, the class of oriented matroids is a minor and dual closed class of matroids.

A circuit $X\in \mathcal{O}$ is positive if $X^{-}=\varnothing$. We say that $\mathcal{O}$ is totally cyclic if each $e\in E$ is contained in a positive circuit $X\in \mathcal{O}$ and that $\mathcal{O}$ is acyclic if no $X\in \mathcal{O}$ is positive. From [14] (Theorem 2.2),

$$\begin{array}{c}\hfill \mathcal{O}\mathrm{is}\mathrm{acyclic}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mathcal{O}}^{*}\mathrm{is}\mathrm{totally}\mathrm{cyclic}.\end{array}$$

(10)

For any $Z\subseteq E$, denote by ${}_{\overline{Z}}M$ the oriented matroid obtained from M by reversing signs on Z, i.e., ${}_{\overline{Z}}M=(E,{}_{\overline{Z}}\mathcal{O})$, where ${}_{\overline{Z}}\mathcal{O}=\{{}_{\overline{Z}}X;X\in \mathcal{O}\}$, supposing that ${}_{\overline{Z}}X$ satisfies ${\left({}_{\overline{Z}}X\right)}^{+}=(X^{+}\setminus Z)\cup (X^{-}\cap Z)$ and ${\left({}_{\overline{Z}}X\right)}^{-}=(X^{-}\setminus Z)\cup (X^{+}\cap Z)$. Set ${\chi }_{Z,E}:E\to \{1,-1\}$ such that ${\chi }_{Z,E}\left(x\right)=-1$ if $x\in Z$ and ${\chi }_{Z,E}\left(x\right)=1$ if $x\in E\setminus Z$. If X is a directed circuit of $\mathcal{O}$ with the signature ${\mathrm{sg}}_X$, then the signature of ${}_{\overline{Z}}X$ ($\in {}_{\overline{Z}}\mathcal{O}$) satisfies ${\mathrm{sg}}_{{}_{\overline{Z}}X}\left(x\right)={\mathrm{sg}}_X\left(x\right){\chi }_{Z,E}\left(x\right)$ for each $x\in \underline{X}$. Thus, ${\chi }_{Z,E}$ uniquely determines ${}_{\overline{Z}}M$.

Let M be an oriented matroid on E and $e\in E$. From [15] (Lemma 3.1.1),

$$\begin{array}{c}\hfill \mathrm{if}\mathrm{both}M\mathrm{and}{}_{\overline{e}}M\mathrm{are}\mathrm{acyclic},\mathrm{then}\mathrm{both}M-e\mathrm{and}M/e\mathrm{are}\mathrm{acyclic};\end{array}$$

(11)

$$\begin{array}{c}\hfill \mathrm{if}M\mathrm{is}\mathrm{acyclic}\mathrm{and}{}_{\overline{e}}M\mathrm{is}\mathrm{not}\mathrm{acyclic},\mathrm{then}M-e\mathrm{is}\mathrm{acyclic}\mathrm{and}M/e\mathrm{is}\mathrm{not}\mathrm{acyclic};\end{array}$$

(12)

$$\begin{array}{c}\hfill \mathrm{if}e\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{loop}\mathrm{of}M\mathrm{and}\mathrm{both}M\mathrm{and}{}_{\overline{e}}M\mathrm{are}\mathrm{not}\mathrm{acyclic},\mathrm{then}\mathrm{both}M-e\mathrm{and}M/e\mathrm{are}\mathrm{not}\mathrm{acyclic}.\end{array}$$

(13)

Suppose that $\mathcal{M}$ is a B-class of oriented matroids. For any $M\in \mathcal{M}$ and $Y\subseteq B$, denote by ${\mathcal{A}}_{M,Y}$ the set of subsets Z of $E\setminus B$ such that ${}_{\overline{Z\cup Y}}M$ is acyclic. Since ${}_{\overline{Z\cup Y}}M$ is uniquely determined by ${\chi }_{Z\cup Y,E\left(M\right)}$, ${\mathcal{A}}_{M,Y}$ can be considered a set of $\{1,-1\}$-functions ${\chi }_{Z\cup Y,E\left(M\right)}$ corresponding to acyclic orientations. Denote by $\mathcal{A}$ the union of ${\mathcal{A}}_{M,Y}$, where M runs through $\mathcal{M}$ and Y runs through the subsets of B. We claim that $\mathcal{A}$ is $(2,0,1,1)$-regular.

Lemma 4.

For any $M\in \mathcal{M}$, $e\in E\setminus B$, and $Y\subseteq B$,

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{A}}_{M,Y}|=2|{\mathcal{A}}_{M-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}M,\hfill \\ |{\mathcal{A}}_{M,Y}|=0|{\mathcal{A}}_{M-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}M,\hfill \\ |{\mathcal{A}}_{M,Y}|=|{\mathcal{A}}_{M-e,Y}|+|{\mathcal{A}}_{M-e,Y}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

Proof. 

The statement is obvious if e is an isthmus or a loop of M. Let $e\in E$ be neither an isthmus nor a loop of M. For a subset Z of $E\setminus B$, set $f(M;Z)=0$ if ${}_{\overline{Z\cup Y}}M$ is not acyclic and $f(M;Z)=1$ if ${}_{\overline{Z\cup Y}}M$ is acyclic. We have

$$\begin{array}{c}\hfill |{\mathcal{A}}_{M,Y}|=\sum _{Z\subseteq (E\setminus B)}f(M;Z).\end{array}$$

If $e\in E\setminus B$ is not a loop of M and Z is a subset of $(E\setminus B)\setminus \left\{e\right\}$, then from (11)–(13), we have

$$\begin{array}{c}\hfill f(M;Z)+f({}_{\overline{e}}M;Z)=f(M-e;Z)+f(M/e;Z).\end{array}$$

Now, $f({}_{\overline{e}}M;Z)=f(M;Z\cup \left\{e\right\})$. Summing up for all subsets Z of $(E\setminus B)\setminus \left\{e\right\}$, we obtain $|{\mathcal{A}}_{M,Y}|=|{\mathcal{A}}_{M-e,Y}|+|{\mathcal{A}}_{M-e,Y}|$ as required. □

Considering $\mathcal{A}$ as the class of $\{1,-1\}$-functions on matroids from $\mathcal{M}$ corresponding to acyclic orientations, any $g:B\to \{1,-1\}$ coincides with $Y\subseteq B$ such that $g={\chi }_{Y,B}$. Thus, we can write ${\mathcal{A}}_{M,Y}$ and ${\mathsf{\chi }}_{n,Y}$ instead of ${\mathcal{A}}_{M,g}$ and ${\mathsf{\chi }}_{n,g}$, respectively. We apply this notation in the following corollary of Theorem 1.

Corollary 4.

Suppose that $\mathcal{M}$ is a B-class of oriented matroids, with B finite, and let $\mathcal{A}$ be the class of acyclic orientations of oriented matroids from $\mathcal{M}$. Assume that ${\mathcal{M}}_{B,\mathcal{A}}=(M_1,\cdots ,M_n)$. Then, for each $M\in \mathcal{M}$, there exists an integral vector ${\mathbf{x}}_M=(x_1,\cdots ,x_n)$ such that for every $Y\subseteq B$, $|{\mathcal{A}}_{M,Y}|={\mathsf{\chi }}_{n,Y}·{\mathbf{x}}_M$.

Proof. 

It follows from Lemma 4 and Theorem 1. □

For any $Y\subseteq B$, denote by ${\mathcal{C}}_{M,Y}$ the set of subsets Z of $E\setminus B$ such that ${}_{\overline{Z\cup Y}}M$ is totally cyclic. Since ${}_{\overline{Z\cup Y}}M$ is uniquely determined by ${\chi }_{Z\cup Y,E\left(M\right)}$, ${\mathcal{C}}_{M,Y}$ can also be considered set of $\{1,-1\}$-functions ${\chi }_{Z\cup Y,E\left(M\right)}$ corresponding to totally cyclic orientations. Denote by $\mathcal{C}$ the union of ${\mathcal{C}}_{M,Y}$, where M runs through $\mathcal{M}$ and Y runs through the subsets of B. We claim that $\mathcal{C}$ is $(0,2,1,1)$-regular.

Lemma 5.

For any $M\in \mathcal{M}$, $e\in E\setminus B$, and $Y\subseteq B$,

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{C}}_{M,Y}|=0|{\mathcal{C}}_{M-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}M,\hfill \\ |{\mathcal{C}}_{M,Y}|=2|{\mathcal{C}}_{M-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}M,\hfill \\ |{\mathcal{C}}_{M,Y}|=|{\mathcal{C}}_{M-e,Y}|+|{\mathcal{C}}_{M-e,Y}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

Proof. 

It follows from Lemma 4 and (10). □

Similar to the above, $\mathcal{C}$ can be considered the class of $\{1,-1\}$-functions on matroids from $\mathcal{M}$ corresponding to totally cyclic orientations. Any $g:B\to \{1,-1\}$ coincides with $Y\subseteq B$ such that $g={\chi }_{Y,B}$, and we can write ${\mathcal{C}}_{M,Y}$ and ${\mathsf{\chi }}_{n,Y}$ instead of ${\mathcal{C}}_{M,g}$ and ${\mathsf{\chi }}_{n,g}$, respectively.

Corollary 5.

Suppose that $\mathcal{M}$ is a B-class of oriented matroids, with B finite, and let $\mathcal{C}$ be the class of totally cyclic orientations of oriented matroids from $\mathcal{M}$. Assume that ${\mathcal{M}}_{B,\mathcal{C}}=(M_1,\cdots ,M_n)$. Then, for each $M\in \mathcal{M}$, there exists an integral vector ${\mathbf{x}}_M=(x_1,\cdots ,x_n)$ such that for every $Y\subseteq B$, $|{\mathcal{C}}_{M,Y}|={\mathsf{\chi }}_{n,Y}·{\mathbf{x}}_M$.

Proof. 

It follows from Corollaries 4 and (10). □

Let M be a regular matroid on E associated with a totally unimodular matrix D and N be the regular chain group associated with D. The set of circuits of M coincides with the set of supports of primitive chains of N. If fact, each circuit $C\subseteq E$ of M corresponds to exactly one primitive function $f_C$ of N such that $\sigma \left(f_C\right)=C$. The set of primitive functions forms a set of oriented circuits of an oriented matroid (see [15]). Thus, we can apply Lemmas 4 and 5 and Corollaries 4 and 5 for any B-class of regular matroids.

6. Orientations of Graphs

Consider a fixed orientation D of a graph G. Each circuit C in G indicates two directed circuits; we denote one of them by Q and the other one by $Q^{-1}$. The edges of C and Q indicate a signed set X such that $\underline{X}=E\left(C\right)$, $X^{+}$ consists of the edges having the same orientation in D and Q, and $X^{-}$ consists of the edges having different orientations in D and Q. Then, $Q^{-1}$ indicates $-X$ in an analogous way. Applying this process for each circuit of G, we generate a set $\mathcal{O}$ such that $\left(E\right(G),\mathcal{O})$ is an oriented matroid M on $E\left(G\right)$, and the underlying matroid $\underline{M}$ is the cycle matroid of G; i.e., $\underline{\mathcal{O}}$ is the set of circuits of G.

If $Z\subseteq E\left(G\right)$, then denote by ${}_{\overline{Z}}D$ the orientation of G arising from D after changing the orientation of edges from Z. Clearly, ${}_{\overline{Z}}D$ corresponds to ${}_{\overline{Z}}M$. Analogously, an orientation D of G is totally cyclic if each edge of G is covered by a directed circuit and is acyclic if no edge of G is covered by a directed circuit.

Recall that a B-class of graphs is a class $\mathcal{G}$ such that for each $G\in \mathcal{G}$, $B\subseteq E\left(G\right)$, and for each $e\in E\left(G\right)\setminus B$, $G-e,G/e\in \mathcal{G}$. The class $\mathcal{A}$ (resp. $\mathcal{C}$) of acyclic (resp. totally cyclic) orientations of digraphs from $\mathcal{G}$ is the class of acyclic (resp. totally cyclic) orientations of matroids from the class of cyclic matroids of graphs from $\mathcal{G}$. Similarly, we write ${\mathcal{A}}_{G,Y}$ (resp. ${\mathcal{C}}_{G,Y}$) instead of ${\mathcal{A}}_{M,Y}$ (resp. ${\mathcal{C}}_{M,Y}$), supposing that M denotes the cyclic matroid of G. Analogously, we write ${\mathcal{G}}_{B,\mathcal{A}}=(G_1,\cdots ,G_n)$ instead of ${\mathcal{M}}_{B,\mathcal{A}}=(M_1,\cdots ,M_n)$, where $M_i$ is the cycle matroid of $G_i$ for $i=1,\cdots ,n$.

Lemma 6.

For each $G\in \mathcal{M}$, $e\in E\left(G\right)\setminus B$, and $Y\subseteq B$,

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{A}}_{G,Y}|=2|{\mathcal{A}}_{G-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}G,\hfill \\ |{\mathcal{A}}_{G,Y}|=0|{\mathcal{A}}_{G-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}G,\hfill \\ |{\mathcal{A}}_{G,Y}|=|{\mathcal{A}}_{G/e,Y}|+|{\mathcal{A}}_{G-e,Y}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

(14)

Proof. 

It follows from Lemma 4. □

Corollary 6.

Suppose that $\mathcal{B}$ is a B-class of graphs, with B finite, and let $\mathcal{A}$ be the class of acyclic orientations of graphs from $\mathcal{G}$. Assume that ${\mathcal{G}}_{B,\mathcal{A}}=(G_1,\cdots ,G_n)$. Then, for each $G\in \mathcal{G}$, there exists an integral vector ${\mathbf{x}}_G=(x_1,\cdots ,x_n)$ such that for every $Y\subseteq B$, $|{\mathcal{A}}_{G,Y}|={\mathsf{\chi }}_{n,Y}·{\mathbf{x}}_G$.

Proof. 

It follows from Corollary 4. □

Lemma 7.

For each $G\in \mathcal{M}$, $e\in E\left(G\right)\setminus B$, and $Y\subseteq B$,

$$\begin{array}{c}\hfill \begin{array}{cc}|{\mathcal{C}}_{G,Y}|=0|{\mathcal{C}}_{G-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathit{an}\mathit{isthmus}\mathit{of}G,\hfill \\ |{\mathcal{C}}_{G,Y}|=2|{\mathcal{C}}_{G-e,Y}|\hfill & \mathit{if}e\mathit{is}\mathrm{a}\mathit{loop}\mathit{of}G,\hfill \\ |{\mathcal{C}}_{G,Y}|=|{\mathcal{C}}_{G/e,Y}|+|{\mathcal{C}}_{G-e,Y}|\hfill & \mathit{otherwise}.\hfill \end{array}\end{array}$$

(15)

Proof. 

It follows from Lemma 5. □

Corollary 7.

Suppose that $\mathcal{B}$ is a B-class of graphs, with B finite, and let $\mathcal{C}$ be the class of totally cyclic orientations of graphs from $\mathcal{G}$. Assume that ${\mathcal{G}}_{B,\mathcal{C}}=(G_1,\cdots ,G_n)$. Then, for each $G\in \mathcal{G}$, there exists an integral vector ${\mathbf{x}}_G=(x_1,\cdots ,x_n)$ such that for every $Y\subseteq B$, $|{\mathcal{C}}_{G,Y}|={\mathsf{\chi }}_{n,Y}·{\mathbf{x}}_G$.

Proof. 

It follows from Corollary 5. □