Matroids are an important generalization of graphs and matrices; graphic and representable matroids have been the focus of much of the text. But other important combinatorial structures also have interpretations as matroids. In this chapter, we concentrate on two well-studied applications: transversalmatroids (which arise from bipartite graphs) and hyperplane arrangements in ℝn, which are closely related to representable matroids.

Transversal matroids

Finding matchings in bipartite graphs is an extremely important and well-studied topic in combinatorics. For instance, the job assignment problem introduced in Example 1.20 in Chapter 1 asks you to determine which applicants to hire for a collection of jobs. This motivates the notion of a transversal matroid, a matroid associated with matchings in a bipartite graph. These matroids were defined in Example 1.20, and Theorem 7.2 asserts the collections of vertices that can be matched to satisfy the independent set axioms. In Chapter 1, we postponed the proof until “later.” Now, it's later.

In an effort to make this chapter self-contained (and to spare you the trouble of leafing back to Chapter 1), we remind you of all the relevant definitions. Let G be a bipartite graph with vertex bipartition V = X ∪ Y. Recall a subset of edges N is called a matching if no two edges ofN share any vertex. We writeN = (I, J) for I ⊆ X and J ⊆ Y for the vertices that the matching N uses in X and Y, respectively.

Matroids – a quick prehistory

Matroid theory is an active area of mathematics that uses ideas from abstract and linear algebra, geometry, combinatorics and graph theory. The study of matroids thus offers students a unique opportunity to synthesize several different areas within mathematics typically studied at the undergraduate level. Furthermore, matroids are an active area of research; Mathematical Reviews lists some 2000 publications with the word “matroid” in the title, with more than a third of these appearing in the last decade.

Why have we written this book? Our motivation is direct: There is no comprehensive text written for undergraduates on this topic. There are several more advanced treatments of the subject, suitable for graduate students or researchers, butmost of these are difficult for undergraduates to read. To paraphrase an old joke, this text seeks to fill this “muchneeded gap.”

This text introduces matroids by emphasizing geometry, focusing especially on geometric (affine) dependence. Interpreting this approach for finite subsets of a vector space, points in Euclidean space or the edges of a graph gives a matroid spin to linear algebra, discrete geometry and graph theory. We believe the geometric approach, which both authors learned from their common Ph.D. advisor, Thomas Brylawski, to be the most natural, useful and powerful in understanding the subject.

The common thread that ties the various classes of matroids together is the abstract notion of independence. This unifying idea is due to Hassler Whitney, who defined matroids in his foundational paper [42] in 1935.

Matrices are matroids

In Whitney's foundational paper [42], matroids were introduced as an abstraction of linear dependence. Throughout the history of the subject, connections between matrices and matroids have motivated an enormous amount of research. The first three questions we consider are fundamental:

1. Q1. Does every subset of vectors give rise to a matroid, i.e., do the subsets of linearly independent column vectors of a matrix always satisfy the independent set axioms? (Answer: Yes – Theorem 6.1.)

2. Q2. When do two different matrices give the same matroid? (Answer: Sometimes – Section 6.2.)

3. Q3. Does every matroid arise from the linear dependences of some collection of vectors? (Answer: No – Example 6.20.)

Concerning Q1, we've seen plenty of examples of matroids that come from specific matrices so far. In Chapter 1, matrices were the first examples we considered, and, in Chapter 3, we interpreted the matroid operations of deletion, contraction and duality for matrices. But, if you've been paying close attention, we never proved the subsets of linearly independent vectors satisfy the axioms (I1), (I2) and (I3). We fix that now by giving a proof that linear independence satisfies the matroid independence axioms.

Column dependences of a matrix

Theorem 6.1. Let E be the columns of a matrix A with entries in a field F, and let I be those subsets of E that are linearly independent. Then I is the family of independent sets of a matroid.

Matroids can be thought of in many different ways; we tried to make that point in Chapter 2. But the common thread running through all of our different approaches to the subject is the underlying connection to geometry. When we “draw a picture of amatroid,” we are thinking of the elements of the matroid as points and the dependences as lines, planes, and so on.

Geometry in the plane motivates our treatment of affine geometry. Although the word “affine” may be unfamiliar, affine geometry based on coordinates covers very familiar material; points in the plane correspond to ordered pairs (x, y), points in three-dimensions correspond to ordered triples (x, y, z), and so on. Lines in the plane are given by linear equations of the form ax + by = c for constants a, b and c, planes in 3-space are described by equations of the form ax + by + cz = d, and this also generalizes to higher dimensions.

From the geometric point of view, here's what you learned long ago about points and lines in the plane:

1. A: Every pair of points determines a unique line, and

2. B: Given a point P and a line l not containing P, there is a unique line through P parallel to l.

Our matroid interpretation for property A is direct;

• If a and b are non-parallel points in a matroid, then they determine a unique rank 2 flat of the matroid – see Figure 5.1.

Motivation and history

It's not unusual for an open problem in mathematics to motivate signifi- cant research. One of the touchstone problems of modern combinatorics and graph theory was the Four Color Problem, now the Four Color Theorem after its resolution by Appel and Haken in 1976 [1]. This problem, which dates to a letter Francis Guthrie wrote to his brother Frederick in 1852, asks if it is always possible to color the regions of a map with four (or fewer) colors so that no two adjacent regions receive the same color. This is obviously important when you are looking at a map – you don't want regions (countries or states) that share a border to receive the same color.

Appel and Haken's proof was noteworthy for two reasons:

• It resolved (in the affirmative) a 125 year-old conjecture.

• It was the first significant mathematical proof that made essential use of a computer.

In fact, Appel and Haken needed more than 1000 hours of computer time to complete the case checking involved in their proof. A more recent proof [28], modeled on the same approach, but streamlined, dramatically reduces the number of cases to check. But even this new proof uses a computer in an essential way to check cases.

Although the original problem is phrased in terms of coloring regions in a map, you can turn the entire enterprise into coloring vertices of a graph.

Chapter 1 provided an introduction to matroids via a glimpse at several prototypes and their properties. This chapter will give the formal definitions and some of the basic results we will need for the rest of the book.

One of the most attractive features of matroids is the plethora of equivalent definitions. Borrowing from linear algebra, we defined a matroid in terms of independent sets in Chapter 1. In this chapter, we continue in this vein and give an equivalent definition in terms of bases. Graph theory motivates alternate formulations of matroids in terms of circuits and cocircuits, while closed sets, closure and hyperplanes come from geometry. We will tie these disparate concepts together using the rank function. There are many additional ways to define a matroid; we'll discuss some of them in the chapter and the exercises.

Each formulation (or reformulation) of the definition of a matroid will require a finite, ground set E and a family of subsets of E or a function defined on the power set 2E. These subsets and functions will satisfy various axioms, and each set of axioms will have one or two easy conditions and one more substantial axiom that characterizes a matroid property.

The central idea of this chapter is that these formulations are logically equivalent, and it is instructive to explicitly convert one axiom system to another. For example, you might wonder how you could determine the independent sets from the bases, or – if you were told the rank of any subset of the matroid, but nothing else – could you still figure out the independent sets? We'll describe a method, called a cryptomorphism, for converting from one system to another.

Given a matroid, how can you use that matroid to create a new matroid? When does a given matroid (or a vector space, or a group, or a ring, …) decompose as a union or a sum of two (or more) smallermatroids? These are important questions, and this chapter attempts to answer them. We beginwith two very important operations on amatroid that aremotivated by graph theory.

Matroid deletion and contraction

This section is devoted to two very important operations we can perform on a matroid: deletion and contraction. Both operations reduce the size of the matroid by removing an element from M; this is frequently useful for proofs involving mathematical induction. Recall that an element e of a matroid is an isthmus if it's in every basis, and e is a loop if it's in no basis. (For a quick review of the different ways to describe an isthmus and a loop, see Exercises 6 and 7 from Chapter 2.)

Definition 3.1. Let M be a matroid on the ground set E with independent sets I.

Deletion

(1) Deletion For e ∈ E (e not an isthmus), the matroid M − e has ground set E − {e} and independent sets that are those members of I that do not contain e:

I is independent in M − e if and only if e ∉ I and I is independent in M.