Manoel Lemos

A sharp bound on the size of a connected matroid

This paper proves that a connected matroid M in which a largest circuit and a largest cocircuit have c and c* elements, respectively, has at most 2 cc* elements. It is also shown that if e is an element of M and ce and c* are the sizes of a largest circuit containing e and a largest cocircuit containing e, then IE(M)I < (Ce-1)(Ce -1)+1. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehmans width-length inequality which asserts that the former inequality can be reversed for regular matroids when ce and c* are replaced by the sizes of a smallest circuit containing e and a smallest cocircuit containing e. Moreover, it follows from the second inequality that if u and v are distinct vertices in a 2-connected loopless graph G, then IE(G)l cannot exceed the product of the length of a longest (u, v)-path and the size of a largest minimal edge-cut separating u from v.

On removable circuits in graphs and matroids

We shall prove that for any graph H that is a core, if χ(G) is large enough, then H × G is uniquely H-colorable. We also give a new construction of triangle free graphs, which are uniquely n-colorable.

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

Abstract 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 M⧹e 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) .

On packing minors into connected matroids

Abstract Let N be a matroid with k connected components and M be a minor-minimal connected matroid having N as a minor. This note proves that |E(M) − E(N)| is at most 2k − 2 unless N or its dual is free, in which case |E(M) − E(N)| ⩽ k − 1. Examples are given to show that these bounds are best possible for all choices for N. A consequence of the main result is that a minimally connected matroid of rank r and maximum circuit size c has at most 2r − c + 2 elements. This bound sharpens a result of Murty.

Non-separating cocircuits in matroids

In this note, we obtain a lower bound for the number of connected hyperplanes of a 3-connected binary matroid M containing a fixed set A provided M|A is coloopless.

On the 3-connected matriods that are minimal having a fixed spanning restriction

Abstract Let N be a minor of a 3-connected matroid M and let M′ be a 3-connected minor of M that is minimal having N as a minor. This paper commences the study of the problem of finding a best-possible upper bound on |E(M′)−E(N)| . The main result solves this problem in the case that N and M have the same rank.

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K5\ e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F7 with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in P9. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F7 and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R10, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).

Obstructions to a binary matroid being graphic

Bixby and Cunningham showed that a 3-connected binary matroid M is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid M is graphic if and only if it has exactly r(M)+1 non-separating cocircuits. Hence the presence in M of either an element in at least three non-separating cocircuits, or of at least r(M)+2 non-separating cocircuits, implies that M is non-graphic. We provide lower bounds on the size of the set of such elements, and on the number of non-separating cocircuits, in such non-graphic binary matroids. A computationally efficient method for finding such lower bounds for specific minor-closed classes of matroids is given. Applications of this method and other results on sets of obstructions to a binary matroid being graphic are given.

Matroids having the same connectivity function

Abstract We define the connectivity function of a matroid M on a set E as ξ(M;X)=r(X)+r(E⧹X)−r(E)+1(X⊆E), where r is the rank function of M. Cunningham conjectured that a connected matroid is determined, up to duality, by its connectivity function. Seymour (1988) proved this for binary matroids and we shall prove it in general provided r(M)≠r(M ∗ ) . Seymour in 1988 gave a counterexample for this conjecture.

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.