Towards Unavoidable Minors of Binary 4-connected Matroids

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

Towards Unavoidable Minors of Binary 4-connected Matroids

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

Maintaining Matroid 3-Connectivity With Respect to a Fixed Basis

<p>We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M, there exists a set K [is a subset of] E(M) of four distinct elements such that for all k [is an element of the set] K, si(M=k) is 3-connected whenever k [is an element of the set] B, and co(M\k) is 3-connected whenever k [is an element of the set] E(M) - B. Moreover, we show that if no other elements of E(M) - K satisfy this property, then M necessarily has path-width 3.</p>

Maintaining Matroid 3-Connectivity With Respect to a Fixed Basis

<p>We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M, there exists a set K [is a subset of] E(M) of four distinct elements such that for all k [is an element of the set] K, si(M=k) is 3-connected whenever k [is an element of the set] B, and co(M\k) is 3-connected whenever k [is an element of the set] E(M) - B. Moreover, we show that if no other elements of E(M) - K satisfy this property, then M necessarily has path-width 3.</p>

The connectivity and Hamiltonian properties of second-order circuit graphs of wheel cycle matroids

Let [Formula: see text] be the [Formula: see text]th-order circuit graph of a simple connected matroid M. The first-order circuit graph is also called a circuit graph. There are lots of results about connectivity and Hamiltonian properties of circuit graph of matroid, while there are few related results on the second-order circuit graph of a matroid. This paper mainly focuses on the connectivity and Hamiltonian properties of the second-order circuit graphs of the cycle matroid of wheels. It determines the minimum degree and connectivity of these graphs, and proves that the second-order circuit graph of the cycle matroid of a wheel is uniformly Hamiltonian.

Elastic Elements in 3-Connected Matroids

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $\textrm{co}(M\backslash e)$, the cosimplification of $M\backslash e$, or $\textrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $\textrm{co}(M\backslash e)$ and $\textrm{si}(M/e)$ are $3$-connected. Calling such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans.

On the Ehrhart Polynomial of Minimal Matroids

We provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design and Coding

In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the (M,v) almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order n≤5 with the exception of log10·U2,5 and logv·U3,5 for some v.