Let M be a finite matroid with rank function r. We will write A ⊆ M when we mean that A is a subset of the ground set of M , and write M | A and M/A for the matroids obtained by restricting M to A, and contracting M on A respectively. Let M * denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial T M (x, y) satisfies (1) T M (x, y) = A⊆M T M...

A matroid M is a finite se t M of e le me nts with a famil y of subsets, called independent, such th a t (I) every subset of an independe nt se t is independent, and (2) for e ve ry subset A of M , all maximal indepe nde nt s ubsets of A have the sa me cardinality , called the rank r\A) of A. It is proved that a matroid can be partitioned into as few as k sets , each inde pende nt , if and only...

Branch-width is a structural parameter very closely related to tree-width, but branch-width has an immediate generalization from graphs to matroids. We present an algorithm that, for a given matroid M of bounded branch-width t which is represented over a finite field, finds a branch decomposition of M of width at most 3t in cubic time. Then we show that the branch-width of M is a uniformly fixe...

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+ 1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most r rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid inM can be covered by at most cq rank-a sets. In the la...

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh’s 1969 conjecture, which provides a progressive lower bound for t...

A cycle in a matroid is a disjoint union of circuits. This paper proves that every regular matroid M without coloops has a set S of cycles whose union is E(M) such that every element is in at most 3 of the cycles in S. It follows immediately from this that, on average, each element of M is in at most 3 members of the cycle cover S. This improves on a 1989 result of Jamshy and Tarsi who proved t...

Let $G=(V, E)$ be an undirected graph. The spanning trees polytope $P(G)$ is the convex hull of characteristic vectors all $G$. In this paper, we describe facets as a consequence bases $P(M)$ matroid $M$, i.e., $M$.

Tutte defined a k–separation of a matroid M to be a partition (A, B) of the ground set of M such that |A|, |B| ≥ k and r(A) + r(B) − r(M) < k. If, for all m < n, the matroid M has no m–separations, then M is n–connected. Earlier, Whitney showed that (A, B) is a 1–separation of M if and only if A is a union of 2–connected components of M . When M is 2–connected, Cunningham and Edmonds gave a tre...

Las Vergnas’ generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariantsfis modified so the Tutte decomposition ,f(M) =f(M\e) +f( M/e) is applied only when e $ P (and e is neither a loop nor an isthmus) where P is a distinguished set of points called ports. The resulting “P-ported” Tutte polynomial tp has variables I, w; q,, qZ, . . . . qm...

We consider a specialization YM (q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M . We show that the coefficients of YM (1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independe...