On minor-minimally-connected matroids

AbstractBy a well-known result of Tutte, if e is an element of a connected matroid M, then either the deletion or the contraction of e from M is connected. If, for every element of M, exactly one of these minors is connected, then we call M minor-minimally-connected. This paper characterizes such matroids and shows that they must contain a number of two-element circuits or cocircuits. In addition, a new bound is proved on the number of 2-cocircuits in a minimally connected matroid

On a matroid identity

The purpose of this note is to prove an identity for generalized Tutte-Grothendieck invariants, at least two special cases of which have already proved to be of considerable use. In addition, one of these special cases is used to strengthen results of Lindström on the critical exponent of a representable matroid and the chromatic number of a regular matroid. © 1983

On a packing problem for infinite graphs and independence spaces

In this paper several infinite extensions of the well-known results for packing bases in finite matroids are considered. A counterexample is given to a conjecture of Nash-Williams on edge-disjoint spanning trees of countable graphs, and a sufficient condition is proved for the packing problem in independence spaces over a countably infinite set. © 1979

On ternary transversal matroids

AbstractThe purpose of this paper is to answer a question of Ingleton by characterizing the class of ternary transversal matroids

On singleton 1-rounded sets of matroids

The aim of this paper is to show that there are exactly eight connected matroids N with the property that if M is a connected matroid having N as a minor and x is an element of M, then M has a minor isomorphic to N which contains x in its ground set. © 1984

On the Numbers of Bases and Circuits in Simple Binary Matroids

Quirk and Seymour have shown that a connected simple graph has at least as many spanning trees as circuits. This paper extends and strengthens their result by showing that in a simple binary matroid M the quotient of the number of bases by the number of circuits is at least 2. Moreover, if M has no coloops and rank r, this quotient exceeds 6(r + 1)/19. © 1983, Academic Press Inc. (London) Limited. All rights reserved

On some extremal connectivity results for graphs and matroids

Dirac and Halin have shown for n = 2 and n = 3 respectively that a minimally n-connected graph G has at least ((n-1)|V(G)|-2n)/(2n-1) vertices of degree n. This paper determines the graphs which are extremal with respect to these two results and, in addition, establishes a similar extremal result for minimally connected matroids. © 1982

A note on Negami\u27s polynomial invariants for graphs

Negami has introduced two polynomials for graphs and proved a number of properties of them. In this note, it is shown that these polynomials are intimately related to the well-known Tutte polynomial. This fact is used, together with a result of Brylawski, to answer a question of Negami. © 1989