A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.

The key to Seymour’s Regular Matroid Decomposition Theorem is his result that each 3-connected regular matroid with no R10or R12-minor is graphic or cographic. We present a proof of this in terms of signed graphs. 2004 Wiley Periodicals, Inc. J Graph Theory 48: 74–84, 2005

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the eeectiveness of these paradigms for two optimization problems on matroids: nding an optimum matroid basis and packing disjoint matroid bases. Applications of th...

This paper presents a theorem concerning a matroid with the parity condition. The theorem provides matroid generalizations of graph-theoretic results of Lewin [3] and Gallai [1]. Let M=(E, F) be a matroid, where E is a finite set of elements and F is the family of independent sets of M (in this paper, we presuppose a knowledge of matroid theory; our standard reference is Welsh [4]). Assume that...

A cycle of a matroid is a disjoint union of circuits. A cycle C of a matroidM is spanning if the rank of C equals the rank ofM . Settling an open problem of Bauer in 1985, Catlin in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44] showed that if G is a 2-connected graph on n > 16 vertices, and if δ(G) > n 5−1, thenGhas a spanning cycle. Catl...

The extension space conjecture of oriented matroid theory claims that the space of all (non-zero, non-trivial, single-element) extensions of a re-alizable oriented matroid of rank r is homotopy equivalent to an (r ? 1)-sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. Th...

In this paper, we present an O(r n) algorithm for the linear matroid parity problem. Our solution technique is to introduce a modest generalization, the non-simple parity problem, and identify an important subclass of non-simple parity problems called 'easy' parity problems which can be solved as matroid intersection problems. We then show how to solve any linear matroid parity problem parametr...

We discuss how to obtain an implicit description of the closure of a discrete exponential family with a finite set of equations derived from an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This framework allows us to study the possible support sets of an exponential families with...

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the eeectiveness of these paradigms for two optimization problems on matroids: nding an optimum matroid basis and packing disjoint matroid bases. Applications of th...

We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space which specified classes of edges are allowed to change length coordinated fashion that requires differences lengths be preserved within each class. Rigidity these is generic property, and we characterize the rigid graphs terms redundant standard $d$-dimensional matroid. also interpret our main results matroid unions.