Matroid Theory

Spring 2013

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This course is an introduction to the basic concepts and constructions of matroid theory and to the chief examples. This is not an introductory graduate course. The absolute minimum requirement is a good understanding of abstraction and a solid modern algebra background (as from a graduate course in that).

The course is based on the textbook:

James Oxley, Matroid Theory, second edition, Oxford University Press, 2011. (Do not use the first edition.)

The book doesn't cover everything – a good thing, if you prefer your book to be less than 1000 pages long – but it is an exceptionally nice graduate textbook. The book will be on reserve in the Reserve Reading Room (on 2-hour loan).

Optional additional reading:

- Oxley's errata for this edition (as of 2011).
- A condensed introduction to matroids by Oxley.
- A possibly valuable supplement is the undergraduate textbook Matroids: A Geometric Introduction by Gary Gordon and Jennifer McNulty, Cambridge University Press, 2012. It may help you to get better intuition.
- For a signed-graphic proof of Tutte's characterization of regular matroids, see Gerards' paper.

We meet on M, W, F at 2:20 in LN-2201. There is also a *student problem session* (sometimes a makeup class or bull session) on *Thursdays, 1:15-2:15, in LN-2201*.

- Special
**makeup classes**will be arranged to allow for:- one missed day on Feb. 22. (Special lecture on cryptography at the Watson School.)
- one missed day on March 1. (I'll be at the AMS meeting in Oxford, Miss., where there is a "Special Session on Connections between Matroids, Graphs, and Geometry");
- three missed days in the first week of April. (I'll be at the conference "Combinatorial Geometries 2013" in Marseilles.)

**makeup dates: 2/7, 2/28, 4/11, 4/18, 4/25, ...**. - Bull sessions (with me present) on some other Thursdays.

Short outline:

- Oxley, Chapters 1-9 (with some omissions) and perhaps a few more sections. Almost all of Chapters 1-4 is fundamental.
- Omitted: Zaslavsky, "The Möbius function and the characteristic polynomial", Chapter 7 of Combinatorial Geometries, ed. by Neil White.

An outline of the initial material from the course is available here, as a PDF file. It doesn't go far or have much detail. It probably has errors. I will be happy to hear of any.

Supplementary reading (randomly chosen):

- Bartlett, "Chordal graphs", from Mathcamp 2011.
- Blair and Peyton, "An introduction to chordal graphs and clique trees", Oak Ridge Nat. Lab. Report ORNL/TM-12203, 1992.

I expect you, the students, to study the material and to work on as many of the exercises as you can. I'll collect written work every week or two and return it with comments, for your edufication. There will not be any tests! I'll meet separately with each student frequently to discuss your progress and any questions you may have.

ASSIGNMENTS: Click here for

I will post them as we find them and as the course progresses. Link to the corrections page.

Here is Oxley's PDF file of corrections for the first edition as of 2005. Some of them are simple errors, but many are improved proofs, updates on conjectures, and so on. See Oxley's homepage for his short introduction to matroid theory.

Here are additional corrections and comments of my own.

Some of these may be relevant to the second edition; I don't know yet!

What is a matroid, really? A matroid is not one thing; it is a complex of properties, no one of them most important. Here is a description of a function, which could be said just as well of a matroid.

A function was not, for Riemann, a mere set of points; still less was it any of its pictorial representations as a graph or a table; and still less a collection of expressions involving algebraic formulas. ... What then, was a function? It was anobject, from which none of its attributes could properly be detached. Riemann saw a function the way chess grandmasters are said to see a game, all at once, as a unified whole, aGestalt.– Prime Obsession by John Derbyshire, 2003, p. 129.

Matroid theory originated as an abstract study of the properties of linear dependence of finite sets of vectors in a vector space. The idea was to forget the details of linear dependence relations and only remember which sets of vectors are linearly dependent. There are some nice axioms to describe a property abstracting linear dependence – let's call it ``dependence'' – that allow one to define (purely abstractly) such basic vector-space properties dimension (we call it ``rank''), linearly closed sets, bases, and much more. The main thing they don't allow one to do is to find the vector space, because the axioms admit examples that cannot possibly come from vectors. A set with a list of dependent subsets (or an equivalent, like a rank function or a list of bases) is called a matroid.

Another origin of matroid theory is in graph theory. There is a natural matroid on a graph: the elements of the matroid are the edges of the graph and the dependent subsets are the edge sets that contain a circuit (a.k.a. cycle or closed path). Many interesting and important graph properties are naturally expressed in terms of matroids. For instance, a spanning tree of a graph is the same as a basis of the graphic matroid. The chromatic polynomial of a graph (this counts proper colorings; it appeared in an attempt to prove the Four-Color Theorem) is another thing that is essentially matroidal, although the connection is less simple. Graph connectivity is closely related to matroids, although here there are some complications.

There are many other sources of matroid theory and examples, including hyperplane arrangements (which have their own analogs of chromatic polynomials), transversals, and whatever.

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