On bipartite restrictions of binary matroids

In a 1965 paper, Erdos remarked that a graph G has a bipartite subgraph that has at least half as many edges as G. The purpose of this note is to prove a matroid analogue of Erdos\u27s original observation. It follows from this matroid result that every loopless binary matroid has a restriction that uses more than half of its elements and has no odd circuits; and, for 2≤k≤5, every bridgeless graph G has a subgraph that has a nowhere-zero k-flow and has more than k-1/k|E(G)| edges. © 2011 Elsevier Ltd

A Note on the Critical Problem for Matroids

Let M be a matroid representable over GF(q) and S be a subset of its ground set. In this note we prove that S is maximal with the property that the critical exponent c(M|S; q) does not exceed k if and only if S is maximal with the property that c(M · S) ≤ k. In addition, we show that, for regular matroids, the corresponding result holds for the chromatic number. © 1984, Academic Press Inc. (London) Limited. All rights reserved

Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5]