Bounding the beta invariant of 3-connected matroids

Abstract

The beta invariant is related to the Chromatic and Tutte Polynomials and has been studied by Crapo [4], Brylawski [2], Oxley [7] and others. Crapo [4] showed that a matroid with at least two elements is connected if and only if its beta invariant is greater than zero. Brylawski [2] showed that a connected matroid has beta invariant one if and only if M is isomorphic to a serial-parallel network. Oxley [7] characterized all matroids with beta invariant two, three and four. In this paper, we first give a best possible lower bound on the beta invariant of 3-connected matroids, then we characterize all 3-connected matroids attaining the lower bound. We also characterize all binary matroids with beta invariant 5, 6, and 7.

Introduction

The notations and terminology in this paper mainly follow Oxley's paper on the beta invariant [7] and his textbook on the matroid [9]. For example, we use ${\mathcal{W}}_r$ and $M({\mathcal{W}}_r)$ to denote the wheel graph and the cycle matroid of the wheel with r spokes, respectively. The whirl of rank r is denoted by ${\mathcal{W}}^r$. A Prism graph is obtained by adding a perfect matching to two vertex-disjoint triangles. We use $Prism+e$ to denote the unique simple graph obtained by adding an edge to the Prism. The uniform matroid $U_{r,n}$, the Fano matroid $F_7$, and the affine geometry $AG(3,2)$ are all well-known matroids and can be found in Oxley [9].

The focus of this paper is the beta invariant of a matroid including its lower bound. For a matroid M, the beta invariant $\beta (M)$ of M was first introduced by Crapo [4] as follows:

$$\beta (M)={(-1)}^{r(M)}\sum _{A\subseteq E(M)}{(-1)}^{|A|}r(A).$$

Here $r(A)$ is the rank of the set A. The beta invariant is related to the Tutte polynomial, which has numerous applications not only in areas of mathematics such as Knot Theory, but also in other subjects such as statistical physics. Crapo [4] showed that the beta invariant satisfies the deletion-contraction formula. That is, if e is neither a loop nor a coloop, then

$$\beta (M)=\beta (M﹨e)+\beta (M/e).$$

Crapo also proved the following interesting result.

Theorem 1.1

[4] i) A matroid M with at least two elements is connected if and only if $\beta (M)>0$.

ii) $\beta (M)=\beta (M^{⁎})$.

iii) If $M_1$ is a series-parallel extension of the loopless matroid M, then $\beta (M_1)=\beta (M)$.

Brylawski's next result characterized all matroids with beta invariant one.

Theorem 1.2

[2] For a connected matroid M on a set of at least two elements, the following statements are equivalent:

(i) $\beta (M)=1$.

(ii) M is a series-parallel extension of $U_{1,1}$.

(iii) M has no minor isomorphic to $U_{2,4}$ or $M(K_4)$.

The next result of Oxley characterized all matroids with beta invariant two, three, and four. The matroid $S_8$ can be found in Oxley [9, page 648]. The geometric representation of non-binary matroids $O_7$, $Q_6$, and $F_7^{-}$ is shown in Fig. 1.

The non-Fano matroid $F_7^{-}$ is obtained from the Fano matroid $F_7$ by relaxing a circuit-hyperplane. $F_7$ has only two binary 3-connected single-element coextensions, one is $AG(3,2)$, and the other is $S_8$. See Oxley [9] for more details.

Theorem 1.3

[7] Let M be a matroid. Then

(i) $\beta (M)=2$ if and only if M is a series-parallel extension of $U_{2,4}$ or $M(K_4)$.

(ii) $\beta (M)=3$ if and only if M is a series-parallel extension of $U_{2,5},U_{3,5},F_7,F_7^{⁎},M({\mathcal{W}}_4)$, or ${\mathcal{W}}^3$.

(iii) $\beta (M)=4$ if and only if either

a) M is a series-parallel extension of one of the matroids $U_{2,6}$, $U_{4,6}$, ${\mathcal{W}}^4$, $M({\mathcal{W}}_5)$, $Q_6$, $O_7$, $O_7^{⁎}$, $F_7^{-}$, ${(F_7^{-})}^{⁎}$, $S_8$, $M(K_5﹨e)$, or $M^{⁎}(K_5﹨e)$, or

b) M is a 2-sum of matroids $M_1$ and $M_2$ each of which is a series-parallel extension of $M(K_4)$ or $U_{2,4}$.

In [1], the authors found all graphs G such that $M(G)$ has beta invariant at most 9. It is an interesting question to bound the beta invariant of a matroid. In the next result, Oxley showed that in general, the beta invariant is bounded below by an exponential function of the connectivity.

Theorem 1.4

[7] For an n-connected matroid M with at least $2(n-1)$ elements, $\beta (M)\ge 2^{n-2}$.

Oxley [7] stated that this bound, in general, seems rather weak. In particular, for 3-connected matroids, the lower bound is only two.

Suppose that $M=M_1{\oplus }_2{M}_2$, the 2-sum of $M_1$ and $M_2$ with the only common element p, where p is neither a loop nor a coloop of $M_1$ or $M_2$, and both $M_1$ and $M_2$ have at least three elements. Oxley [7] showed that

$$\beta (M_1{\oplus }_2{M}_2)=\beta (M_1)\beta (M_2),$$

using Brylawski's result that $\beta (p(M_1,M_2))=\beta (M_1)\beta (M_2)$ [2, Theorem 6.16(vi)]

Proposition 1.5

[7] Let M be a matroid and suppose that $\beta (M)=k>1$. Then either

(i) M is a series-parallel extension of a 3-connected matroid N such that $\beta (N)=k$, or

(ii) $M=M_1{\oplus }_2{M}_2$, and $\beta (M)=\beta (M_1)\beta (M_2)$ with $\beta (M_i)<k$ for $i=1,2$.

As a consequence of the last result, if $\beta (M)$ is prime, then M is a series-parallel extension of a 3-connected matroid N for which $\beta (N)=\beta (M)$.

In this paper, we give a best possible lower bound on the beta invariants of 3-connected matroids with rank r and characterize all matroids that attain the lower bound. The following is our first main result. Our other main results, which can be found in Section 3, will characterize all binary matroids with beta invariants 5, 6, and 7.

Theorem 1.6

Let M be a 3-connected matroid having at least one element and of rank r. Then

• $\beta (M)\ge r(M)-1$. Moreover, $\beta (M)=r(M)-1$ if and only if M is isomorphic to $M({\mathcal{W}}_r)$ ($r\ge 3$), $F_7^{⁎}$, or M(Prism).

• Suppose that M is non-binary. Then $\beta (M)\ge r(M)$; moreover, $\beta (M)=r(M)$ if and only if M is isomorphic to ${\mathcal{W}}^r$, $U_{r,r+2}$ ($r\ge 2$), ${(F_7^{-})}^{⁎}$, or $O_7^{⁎}$.

We use the following results in the proofs of our results. The first result is an extension of Seymour's Splitter Theorem [11].

Theorem 1.7

[3] Let N be a 3-connected proper minor of a 3-connected matroid M such that $|E(N)|\ge 4$ and M is not a wheel or a whirl. Suppose that if $N\cong {\mathcal{W}}^2$, then M has no ${\mathcal{W}}^3$-minor, while if $N\cong M({\mathcal{W}}_3)$, then M has no $M({\mathcal{W}}_4)$-minor. Then M has a 3-connected minor $M_1$ and an element e such that $M_1/e$ or $M_1﹨e$ is isomorphic to N.

Theorem 1.8

[9] (Tutte's Wheels-and-Whirls Theorem) The following are equivalent for a 3-connected matroid M having at least one element.

(i) For every element e of M, neither $M﹨e$ nor $M/e$ is 3-connected.

(ii) M has rank at least three and is isomorphic to a wheel or a whirl.

$R_{10}$ and $R_{12}$ are two important matroids in Seymour's decomposition theory of regular matroids [11]. For details of the matroids, see Oxley [9, Pages 656-657].

Theorem 1.9

[11]

Let M be a 3-connected regular matroid. Then

(i) M is graphic, cographic, or has a minor isomorphic to $R_{10}$ or $R_{12}$.

(ii) If M has a minor isomorphic to $R_{10}$, then $M\cong R_{10}$.

The binary matroid $P_9$ was first defined by Oxley in [8]. Its binary matrix representation is $[I_4|A]$ where A is given next (see Fig. 2).

Proposition 1.10

[8] Every binary 3-connected single-element extension of $S_8$ either is isomorphic to $P_9$ or has an $AG(3,2)$-minor.