A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

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.     

Counterexamples to conjectures on 4-connected matroids

Tutte characterized binary matroids to be those matroids without aU ₄ ² minor. Bixby strengthened Tutte’s result, proving that each element of a 2-connected non-binary matroid is in someU ₄ ² minor. Seymour proved that each pair of elements in a 3-connected non-binary matroid is in someU ₄ ² minor and conjectured that each triple of elements in a 4-connected non-binary matroid is in someU ₄ ² minor. A related conjecture of Robertson is that each triple of elements in a 4-connected non-graphic matroid is in some circuit. This paper provides counterexamples to these two conjectures.     

Bounding the beta invariant of 3-connected matroids

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.     

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

Matroids and Graphs with Few Non-Essential Elements

An essential element of a 3-connected matroid M is one for which neither the deletion nor the contraction is 3-connected. Tutte's Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3-connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3-connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3-connected matroid M for which no single-element contraction is 3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3-connected.     

An Excluded-Minor Problem in Matroids

Consider the class of matroids M with the property that M is not isomorphic to a wheel graph, but has an element e such that both M\e and M/e are isomorphic to a series-parallel extension of a wheel graph. We give a constructive characterization of such matroids by determining explicitly the 3-connected members of the class. We also relate this problem with excluded minor problems.Received May 30, 2003     

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.     

On minor-minimally 3-connected binary matroids

A matroid M is called minor-minimally 3-connected if M is 3-connected and, for each e∈E(M), either M?e or M/e is not 3-connected. In this paper, we prove a chain theorem for the class of minor-minimally 3-connected binary matroids. As a consequence, we obtain a chain theorem for the class of minor-minimally 3-connected graphs.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

On Maximum-sized k-Regular Matroids

 Let k be an integer exceeding one. The class of k-regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank-r regular matroid has the maximum number of points if and only if it is isomorphic to M(K _r+1), the cycle matroid of the complete graph on r+1 vertices. A simple rank-r near-regular matroid has the maximum number of points if and only if it is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3-point line of M(K _r+2) and then contracting this point. This paper determines the maximum number of points that a simple rank-r k-regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such matroid. This matroid is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(K _r+k+1) isomorphic to M(K _k+2) and then contracting each of these points. Revised: July 27, 1998     

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.     

A minor-based characterization of matroid 3-connectivity

It is well known that a matroid is 2-connected if and only if every 2-element set is contained in a circuit, or equivalently, a _U1,2$U_{1,2}$-minor. This paper proves that a matroid is 3-connected if and only if every 4-element set is contained in a minor isomorphic to a wheel of rank 3 or 4; a whirl of rank 2, 3, or 4; or the relaxation of a rank-3 whirl. Some variants of this result are also discussed.     

Rainbow and monochromatic circuits and cocircuits in binary matroids

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if an n-element rank r binary matroid M is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cocircuit. As the class of binary matroids is closed under taking duals, this immediately implies that if M is colored with exactly $n?r$ colors, then M either contains a rainbow colored cocircuit or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.Motivated by a conjecture of Bérczi, Schwarcz and Yamaguchi, we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of its rainbow circuit-free colorings. For simple graphic matroids, we show that there exists a rainbow circuit-free coloring that uses each color at most twice only if the graph is $(2,3)$-sparse, that is, it is independent in the 2-dimensional rigidity matroid. Furthermore, we give a complete characterization of minimally rigid graphs admitting such a coloring.     

Vector representable matroids of given rank with given automorphism group

A simple way of associating a matroid of prescribed rank with a graph is shown. The matroids so constructed are representable over any sufficiency large field. Their use is demonstrated by the following result: Given an integer k?3 and a function G associating a group with each subset of a set S, there is a matroid M(E), representable over any sufficiently large field, such that E ? S, and for any T ?/ S, the rank of M/Tis k, and the automorphine group of MT is isomorphic to G(T).     

Strong Splitter Theorem

The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M ₀, . . . , M _n of 3-connected matroids with ${M_0 \cong N}$ , M _n = M and for ${i \in \{1, \ldots , n}\}$ , M _i is a single-element extension or coextension of M _i?1. Observe that there is no condition on how many extensions may occur before a coextension must occur. We give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, M starting with N and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of thematroids involved is r(M)). Moreover, if two consecutive single-element extensions by elements {e, f} are followed by a coextension by element g, then {e, f , g} form a triad in the resulting matroid.     

Characterizing binary simplical matroids

In an earlier paper we defined a class of matroids whose circuit are combinatorial generalizations of simple polytopes; these matroids are the binary analogue of the simplical geometrics of Crapo and Rota. Here we find necessary and sufficient conditions for a matroid to be isomorphic to such a binary simplical matroid.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.