We prove a sufficient condition for a graph G to have a matching that interconnects all the components of a disconnected spanning subgraph ofG. The condition is derived from a recent extension of the Matroid intersection theorem due to Aharoni and Berger. We apply the result to the problem of the existence of a (spanning) 2-walk in sufficiently tough graphs.

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 nonseparating cocircuits. Hence the presence inM of either an element in at least three non-separating cocircuits, or of at least r(M) + 2 non-separ...

It has been proved by the author that all matroid properties definable in the monadic second-order (MSO) logic can be recognized in polynomial time for matroids of bounded branch-width which are represented by matrices over finite fields. (This result extends so called “MS2-theorem” of graphs by Courcelle and others.) In this work we review the MSO theory of finite matroids and show some intere...

Two simple proofs are given to an earlier partial result about an extremal set theoretic conjecture of Chung, Frank!, Graham, Shearer and Faudree, Schelp, S6s, respectively. The statement is slightly strengthened within a matroid theoretic framework. The first proof re lies on results from matroid theory, while the second is based on an explicit constJuction providing an elementary proof. @ 199...

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that ...

Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of the characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid fa...

We consider the 2-dimensional generic rigidity matroid R(G) of a graph G. The notions of vertex and edge birigidity are introduced. We prove that vertex birigidity of G implies the connectivity of R(G) and that the connectivity of R(G) implies the edge birigidity of G. These implications are not equivalences. A class of minimal vertex birigid graphs is exhibited and used to show that R(G) is no...

Further work of Brylawski and Heron (see [4, p. 315]) explores other characterizations of Eulerian binary matroids. They showed, independently, that a binary matroid M is Eulerian if and only if its dual, M∗, is a binary affine matroid. More recently, Shikare and Raghunathan [5] have shown that a binary matroid M is Eulerian if and only if the number of independent sets of M is odd. This chapte...

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) crypto-morphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. This paper establishes the definition and several basic properties of q-mat...