The 3-connected matroids with circumference 6

We construct all 3-connected matroids with circumference equal to 6 having rank at least 8. A matroid belongs to this family if and only if it is a generalized parallel connection of a set of planes along a common line (which may have some virtual points).     

Connected matroids with the smallest whitney numbers

In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if _X(G) = Σμ(0, x)λ^{r ? r(x)} = Σ (?1)^{r ? k}W_kλ^k is the characteristic polynomial of G, then

$\text{w}{}_{\mathrm{k}}\text{?}\text{r}\text{k}\text{+n}\text{r?1}\text{k}$

Thus γ(G) ? 2^{r ? 1} (n+2), where γ(G) = Σw_k. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then

$\text{w}{}_{\mathrm{i}}\text{?}\text{r}\text{i}\text{+ n}\text{r}\text{i+1}\text{for i>1; w}{}_1\text{?r+n}\text{r}\text{2}\text{? 1;}$

|μ| ? (r? 1)n; and γ ? (2^{r ? 1} ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, $\text{r(G) = r(}\text{G}\text{?}\text{) = 4}$, and $\text{n(G) = n(}\text{G}\text{?}\text{) = 3}$ then $\text{(w}{}_1\text{(G))}{}^4{}_{\mathrm{t-G}}\text{? (w}{}_1\text{(}\text{G}\text{?}\text{)) = (8, 20, 18, 7, 1)}$. Further, if β is the Crapo invariant,

$\text{β(G)=}\text{d}{}_{\mathrm{X}}\text{(G)}\text{dλ}\text{(1)}\text{,}$

then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

The 3-connected binary matroids with circumference 6 or 7

In this paper, we construct all 3-connected binary matroids with circumference equal to 6 or 7 having large rank.     

On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)−5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)−B, then |E(M)|3r(M)−4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)−c(M).     

Supereulerian regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large $n$, every 2-edge-connected simple graph $G$ with $\text{◂=▸}n=\mid V\left(G\right)\mid$ and minimum degree $\text{◂≥▸}\delta \left(G\right)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid $M$, if every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂−▸}r\left(M\right)-5}{5},6\right\}$, then $M$ is supereulerian. We prove the following. (i) Let $M$ be a connected simple regular matroid. If every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂+▸}r\left(M\right)+1}{10},9\right\}$, then $M$ is supereulerian. (ii) For any real number $c$ with , there exists an integer $f\left(c\right)$ such that if every cocircuit $D$ of a connected simple cographic matroid $M$ satisfies $\mid D\mid \ge \text{◂lim▸}\max \text{◂{}▸}\{c\text{◂()▸}\left(r\left(M\right)+1\right),f\left(c\right)\}$, then $M$ is supereulerian.     

Oriented matroids with few mutations

This paper defines a “connected sum” operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ℝP ^d-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ℝP³ with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the “strong-map conjecture” of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.     

Oriented matroids

In this paper, the basic properties of oriented matroids are examined. A topological representation theorem for oriented matroids is proven, utilizing the notion of an “arrangement of pseudo-hemispheres”. The duality theorem of linear programming is extended to oriented matroids.     

On matroids with few circuits containing a pair of elements

In this note, we construct all the matroids that have a pair of elements belonging to just one of its circuits. We use this result to establish that, with two small exceptions, wheels and whirls are the only 3-connected matroids having a pair of elements contained in exactly two of its circuits.     

Bias matroids with unique graphical representations

Given a 3-connected biased graph Ω with three node-disjoint unbalanced circles, at most one of which is a loop, we describe how the bias matroid of Ω is uniquely represented by Ω.     

Biased graphs whose matroids are special binary matroids

Abiased graph is a graph together with a class of polygons such that no theta subgraph contains exactly two members of the class. To a biased graph are naturally associated three edge matroids:G(), L(), L ₀ (). We determine all biased graphs for which any of these matroids is isomorphic to the Fano plane, the polygon matroid ofK ₄,K ₅ orK _3,3, any of their duals, Bixby's regular matroidR ₁₀, or the polygon matroid ofK _m form > 5. In each case the bias is derived from edge signs. We conclude by finding the biased graphs for whichL ₀ () is not a graphic [or, regular matroid but every proper contraction is.Research supported by National Science Foundation grant DMS-8407102 and SGPNR grant 85Z0701Visiting Research Fellow, 1984–1985     

Unipancyclic matroids

Analogous to the concept of uniquely pancyclic graphs, we define a uniquely pancyclic (UPC) matroid of rank $r$ to be a (simple) rank-$r$ matroid containing exactly one circuit of each length $?$ for $3\le ?\le r+1$. Our discussion addresses the existence of graphic, binary, and transversal representations of UPC matroids. Using Shi’s results, which catalogued exactly seven non-isomorphic UPC graphs, we produce a nongraphic binary UPC matroid of rank 24. We consider properties of binary UPC matroids in general, and prove that all binary UPC matroids have a connectivity of $2$.     

Properties of matroids characterizable in terms of excluded matroids

Some properties π of matroids are characterizable in terms of a set S(π) of exluded matroids, that is, a matroid M satisfies property π if and only if M has no minor (series-minor, parallel-minor) isomorphic to a matroid in S(π). This note presents a necessary and sufficient condition for a property to be characterizable in terms of excluded 3-connected matroids.     

Regular matroids with every circuit basis fundamental

Hartvigsen and Zemel have obtained a characterization of those graphs which have every circuit basis fundamental. We consider the corresponding problem for binary matroids. We show that, in general, the class of binary matroids for which every circuit basis is fundamental is not closed under the taking of minors. However, this class is closed under the taking of series-minors. We also describe some general properties of this class of matroids. We end by extending Hartvigsen and Zemel's result to the class of regular matroids, thus obtaining a set of excluded minors which are graphic for this class. © 1996 John Wiley & Sons, Inc.     

Orientable arithmetic matroids

The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of “oriented arithmetic matroid” and prove some properties like the “uniqueness of orientation”.     

Non-regular simplicial matroids

Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.     

Higher support matroids

The purpose of this paper is to provide links between matroid theory and the theory of subcode weights and supports in linear codes. We describe such weights and supports in terms of certain matroids arising from the vector matroids associated to the linear codes. Our results generalize classical results by Whitney, Tutte, Crapo and Rota, Greene, and other authors. As an application of our results, we obtain a new and elegant dual correspondence between the bond union and cycle union cardinalities of a graph.