An upper bound for the circumference of a 3-connected binary matroid

Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid M with an odd circuit, if a largest odd circuit of M has k elements, then a largest circuit of M has at most 2 k − 2 elements. The goal of this note is to show that, when M is 3-connected, either M has a spanning circuit, or a largest circuit of M has at most 2 k − 4 elements. Moreover, the latter holds when M is regular of rank at least four


Introduction
We assume familiarity with matroid theory. Our notation and terminology will follow Oxley [9] except where otherwise indicated. For a positive integer n, we use [n] to denote the set {1, 2, . . . , n}. A circuit C in a matroid is even if |C| is even; otherwise C is odd. A binary matroid is affine if all of its circuits are even. Let M be a matroid having at least one circuit. The circumference, c(M ), of M is the cardinality of a largest circuit of M . If M has an odd circuit, its odd-circumference, c odd (M ), is the cardinality of a largest odd circuit of M . In a private communication to the second author, Jim Geelen and Peter Nelson proved the following result. The proof appears in [10]. The purpose of this note is to prove the next theorem, a refinement of the last result for 3-connected matroids.
Theorem 2. Let M be a 3-connected binary matroid. If M is non-affine, then either M has a spanning circuit, or c(M ) 2c odd (M ) − 4.
As we shall show at the end of Section 4, the bound in the last theorem is sharp for infinitely many ranks. In the next section, we note some preliminary results that will be used in the proof of Theorem 2. The proof of the theorem will be given in Section 3. In Section 4, we show how Theorem 1 can be combined with results concerning matroids of small circumference due to Maia [7], Maia and Lemos [8], and Cordovil, Maia, and Lemos [2] to yield results of Oxley and Wetzler [11] and of Chun, Oxley, and Wetzler [1] up to some small-rank matroids. In Section 5, we conjecture a strengthening of Theorem 2 and we prove this conjecture when M is regular.

Preliminaries
Seymour [12] gave conditions under which a k-separation of a restriction of a matroid could be extended to a k-separation of the whole matroid. In particular, he proved the following result [12, (3.8)].
Theorem 3. Let Z be a set in a matroid M and let (P 1 , P 2 ) be a partition of Z. Then either M/Z has a circuit that is not a circuit of M/P 1 or of M/P 2 , or E(M ) has a partition (X 1 , X 2 ) such that X i ∩ Z = P i for each i in {1, 2} and For a matroid M , recall that C(M ) denotes the set of circuits of M . A subset L of E(M ) is a Tutte-line of M if (M |L) * has rank two and has no loops [13]. As Tutte showed and is easily checked, a Tutte-line L has a partition into sets P 1 , P 2 , . . . , P n for some n 2 such that C(M |L) = {L − P 1 , L − P 2 , . . . , L − P n }. A Tutte-line L is connected if M |L is connected or, equivalently, if n 3.
For a matroid M , a subset S of C(M ) is a linear subclass of circuits of M provided that, for each Tutte-line L of M , either |C(M |L) ∩ S| 1, or C(M |L) ⊆ S. Tutte [13, (4.34)] proved the following result. We shall apply this result here by using the easily verified fact that, when M is binary, the set of all even circuits of M is a linear subclass of circuits of M .
Let C, C 1 , C 2 be as in Lemma 5. If e ∈ C 1 ∩ C 2 , then we say that e is a good chord of C having C 1 and C 2 as its associated circuits. Lemma 6. If C is a circuit of M such that |C| = 4k, then there are circuits C 1 and C 2 of M with C = C 1 C 2 such that |C 1 ∩ C 2 | = 1 and |C 1 | = |C 2 | = 2k + 1.
Proof. As noted above, the set S of even circuits of M is a linear subclass of circuits of M . Choose D ∈ S. By Theorem 4, M has a sequence X 1 , X 2 , . . . , X m of distinct odd circuits with X m = D such that X i−1 ∪ X i is a connected Tutte-line of M for all i in [m], where X 0 = C. Take C 1 = X 1 and C 2 = C 1 C. Then C = C 1 C 2 . As |C| is even and |C 1 | is odd, |C 2 | is odd. The result follows from Lemma 5.
The last lemma can also be proved by applying Lemma 3.2 of [10]. We have presented the proof above to recognize Tutte's contribution to this area. Lemma 7. For a circuit C of M with |C| = 4k, let e be a good chord of C having C 1 and C 2 as its associated circuits. Let A be a C-arc of M and let C A and D A be circuits of M such that C A ∩ D A = A and C A D A = C. Then A crosses e if and only if Moreover, when A crosses e, the circuit space of M |(C ∪ A ∪ e), which is spanned by {C, C 1 , C A }, contains seven non-zero members each of which is the support of a circuit of M |(C ∪ A ∪ e).
Proof. We may assume that e ∈ A otherwise the result follows easily. Thus N * is represented over GF (2) by the matrix whose columns are labelled by the elements of E(N ) and whose rows are the incidence vectors of C 1 , C A , and C. One can now check that the parallel classes of N * are A, {e}, and the non-empty members of P. We deduce that N * is isomorphic to a parallel extension of M (K 4 ) if and only if (2) holds.
Proof. As C A D A = C, the parities of |C A | and |D A | are the same. We may assume that C A and D A are even otherwise the result follows by Lemma 5. Let C 1 and D 1 be the associated circuits of e with respect to C. We may assume that By the last part of Lemma 7, where the last step follows because |C 1 ∩ C A | |D 1 ∩ C A |. As A is non-empty and D is odd, we have a contradiction to the assumption that c odd (M ) = 2k + 1. (i) If S be a series class of N contained in C, then, for each i in [n], there is an (ii) Every series class of N is trivial.
Proof. By Lemma 6, n 1. Clearly the circuit space of N has {C, C 1 , C 2 , . . . , C n } as a basis. We shall first show (i) and use this to deduce (ii).
Let Z be the matrix whose columns are labelled by the elements of E(N ) and whose rows are the incidence vectors of C, X 1 , X 2 , . . . , X n . Then Z represents N * over GF (2). As C ∩ X 1 ∩ X 2 ∩ · · · ∩ X n is non-empty, it is a parallel class of N * . Hence S = C ∩ X 1 ∩ X 2 ∩ · · · ∩ X n , so (i) holds. Now assume that (ii) fails, and let S be a series class of N such that |S| 2. By (i), (3) As e 1 , e 2 , . . . , e n are loops of M/C, it follows that A is a circuit of M/C. Then there are As By (4), (6), and Lemma 7, A crosses {e j }. Thus, by Lemma 8, |A| = 1 and |C A | = |D A | = 2k + 1. If A = {f }, then f is a good chord of C. This is a contradiction as f ∈ {e 1 , e 2 , . . . , e n }. Hence (ii) holds.
Next we prove the main result.
Proof of Theorem 2. We continue to assume that M is a non-affine binary matroid for which c odd (M ) = 2k + 1. In addition, we assume that M is 3-connected and that C is a circuit of M with 4k elements. We shall prove that C spans M . Let e 1 , e 2 , . . . , e n be the good chords of C. By Lemma 6, n 1. For i in [n], let C i and D i be the associated circuits of e i with respect to C. Let N = M |(C ∪ {e 1 , e 2 , . . . , e n }).
We may assume that E(M ) − E(N ) = ∅ otherwise the result holds. Let A be a circuit of M/E(N ) that is not a circuit of M . Note that A is a circuit of M/C because C spans N . Let C A and D A be circuits of M such that the electronic journal of combinatorics 29 (2022), #P00 Suppose that |C A ∩ C| < |A|. Then |D A | = [|C| − |C A ∩ C|] + |A| > |C| = 4k, a contradiction. We conclude that Therefore |C A ∩ C| 2, otherwise |C A ∩ C| = 1, so |A| = 1 and |C A | = 2, a contradiction.
Let a and b distinct elements of C A ∩ C. By Lemma 9(ii), {a} is a series class of N . By Lemma 9(i), for each i in [n], there is an

Consequences
In this section, we note some implications of Theorem 1. We begin with a quick proof of this theorem based on the following 2007 result of Lemos [ Theorem 1 was motivated in part by results of Oxley and Wetzler [11] and Chun, Oxley, and Wetzler [1] determining the simple connected binary matroids with odd-circumference three and the 3-connected binary matroids with odd-circumference five. As Theorem 1 bounds the circumference of a matroid in terms of its odd-circumference, the first of these results can be derived from theorems that determine all simple connected matroids of small circumference. Such matroids having circumference in {3, 4, 5} were found by Maia [7] and Maia and Lemos [8]. We begin with the 3-connected case [8, Theorem 1.2]. The matroid Z 5 is the rank-5 binary spike with tip t.
In [5, Theorem 1.5], we proved the following. These results can now be used to prove the following result of Oxley and Wetzler's [11] that determines all connected binary matroids with odd-circumference three. We denote by K 2,n the graph obtained from K 2,n by adding an edge joining the vertices in the 2-vertex class. This theorem can be combined with Theorem 2 to prove the following result of Chun, Oxley, and Wetzler [1] for matroids of rank at least eight. Note that all of the matroids described in (iii) of this theorem attain the bound in Theorem 2. Their results show that the number c must be even. Moreover, it is possible to construct these matroids keeping c fixed and taking d as large as one desires.

Extensions
We believe that the following extension of Theorem 2 holds.
Conjecture 17. Let M be a 3-connected binary matroid. If M is non-affine, then either M is isomorphic to U 0,1 , U 2,3 , M (K 4 ), or F 7 , or We show next that the conjecture holds when M is regular. Then each column of D has exactly 2k ones. Moreover, since by Lemma 9, all series classes of N are trivial, all of the rows of D are distinct and non-zero. Since k 2, it follows that D has at least seven rows, so D has at least three columns.
For columns a, b, and c of D, let D[a, b, c] be the submatrix of D whose columns are labelled by a, b, and c. Evidently, D[a, b, c] has exactly eight possible different rows. Let x i be the number of rows equal to (t 2 , t 1 , t 0 ) where i = t 2 2 2 + t 1 2 + t 0 . Hence, by permuting rows, the matrix that is obtained from D[a, b, c] by adjoining the column sums a + b + c and a + b + c + z is as shown in Figure 1 where the labels on the rows indicate the multiplicities of the rows and may be zero.
Because each of a, b, and c has exactly 2k ones, we have Thus the electronic journal of combinatorics 29 (2022), #P00 As 7 i=0 x i = 4k − 1, we deduce that We will be interested in the sets C {a, b, c} and C {a, b, c, z} where, for example, the first of these is the union of {a, b, c} and all of the elements of C − z in which the corresponding row of a + b + c is one. Clearly This follows because each column of D has 2k ones and 2k − 1 zeros. 18.3. D[a, b, c] does not have as a submatrix either the matrix F in Figure 2 or any row or column pemutation of F . Assume that D[a, b, c] has F as a submatrix. Observe that a circuit of M contained in C {a, b, c} and containing c must also contain a and b. As C − z is independent, it follows that C {a, b, c} is a circuit of M . Moreover, a circuit of M contained in C {a, b, c, z} and containing z must also contain a and b, and hence c. Thus C {a, b,  Assume that D[a, b] has exactly one row, say row 2k, equal to (1, 1). Then we may assume that the first 2k − 1 rows of D[a, b] equal (1, 0) and the last 2k − 1 rows equal (0, 1). Suppose that D has a column c other than a or b such that its entry in row 2k is equal to one. Then, because D[b, c] has (0, 1) as a row, there must be a one in column c among the first 2k − 1 rows. By symmetry, as D[a, c] has (0, 1) as a row, there is a one in column c among the last 2k − 1 rows. Because c has exactly 2k ones, it must also have a zero among each of its first 2k − 1 and last 2k − 1 rows. It follows that, after a possible row permutation, D[a, b, c] has F as submatrix, a contradiction to 18.3.
We may now assume that, for every column c other than a and b, the entry in row 2k is 0. As c has rows among its first 2k − 1 and last 2k − 1 having entries equal to one, it follows that D[a, b, c] has as a submatrix the 3 × 3 matrix whose rows are (1, 0, 1), (1, 1, 0) and (0, 1, 1). Since z is a column of all ones, it follows that N has the Fano matroid as a minor, a contradiction. We conclude that 18.4 holds.
We may now assume that D[a, b] has at least two rows equal to (1, 1). As the corresponding rows of D are distinct, there is a column c in which the entries in these two rows are distinct. Hence D[a, b, c] has (1, 1, 1) and (1, 1, 0) as rows. Since D[a, c] has (0, 1) as a row, D[a, b, c] has (0, 1, 1) or (0, 0, 1) as a row. Consider the first case. As D[b, c] has (0, 1) as a row, D[a, b, c] has (1, 0, 1) or (0, 0, 1) as a row. The first of these options yields Then [I 4 |D ] represents a minor of M . This minor is the non-regular matroid S 8 , so we have a contradiction that completes the proof.