Math 580A: Matroid Theory
Fall 2023

Corrections and Additions to Oxley, Second Edition

A valuable consequence of Lemma 1.4.6 is the following corollary, which should be stated explicitly:
Corollary 1.4.6.1. If X ⊆ E(M) and C is a circuit with an element y such that C\y ⊆ X, then y ∈ cl(X). In particular, if X is closed, and C\y ⊆ X, then y ∈ X.

Section 1.5 on page 32 describes affine representation of a column matroid, but there is one thing missing from the description. The points of the matroid are given by vectors of size m (in V(m,F)) in a matrix B. When we add the extra coordinate 1, the affine space will be m-dimensional, A(m,F). The matroid will not be the column matroid of B. If we want to represent the column matroid of B by affine points, we will be in A(m−1,F) so we should not add the extra 1's for this purpose.
There are two different ways to think of the columns of a matrix B. (1) We can think of their linear dependence (this gives the column matroid in V(m,F)) or (2) their affine dependence (this gives an affine matroid in A(m,F)). They are not the same matroid. In (1) we can make an affine picture of linear dependence in A(m−1,F). In (2) we get a vector matroid by adding a row of 1's to B.
The conditions 1.5.10-12 on page 38 need to be understood in view of the definitions of matroid point, line, and plane on page 40, which should precede them.

In the middle of page 44, the graphical criterion for a partial transversal is incomplete. It should read: "It is not difficult to see ... in which every edge has an endpoint in X and every vertex in X is an endpoint of an edge."

Problem 2.2.8 on page 84 may have a wrong matrix. In the first edition (Problem 2.2.6) the matrix has a row between our edition's rows 1 and 2, of the form 1,1,0,0,...,0. I recommend inserting this row and solving that matrix. I further recommend solving both matrices and comparing the results. Can you explain the comparison?

Problem 3.2.4 on p. 112 has M \ x,y. That is an error. I think the correction is M/{x,y}. (I'm awaiting verification.)

I (and others) define a graph to be inseparable if it has no cutpoints, including points that separate a loop from the rest of the graph. (That is not the universal definition of a cutpoint.) This is what Oxley calls a block. I use both terms. A single loop and a single link are blocks.

In Sect. 5.3, the term "generalized cycle" should (my opinion) be "circle/cycle of blocks", because it is more precisely descriptive. Definition: A circle of blocks is a graph obtained from blocks B_i, i = 1,...,k (where k ≥ 2), with distinct vertices a_i and b_i, by identifying a_i with b_i−1 for every i. Subscripts are taken modulo k. (Personally, I like the term "necklace of blocks".)
We'll have another use for circles of blocks and Lemma 5.3.4 when discussing frame matroids of biased graphs.

Page 237ff.: The term "unbalanced" should be "contrabalanced" everywhere (including the index). "Unbalanced" means not balanced; a biased graph that contains one unbalanced circle or one half edge is already unbalanced. "Contrabalance" is the complete opposite of balance: no circle is balanced.
Examples: A forest is both balanced and contrabalanced, but not unbalanced. An unbalanced circle is contrabalanced. An unbalanced theta graph may be contrabalanced or not; it is a frame-matroid circuit if and only if it is contrabalanced.

In Sect. 6.10, on page 238, my copy of the book uses the name "bias matroid" after mentioning the alternative name "frame matroid". The name to use is "frame matroid"; "bias matroid" has been abandoned because it causes confusion (and it is not as descriptive, and I never really liked it).

P. 240, line 18: I would add "Biased graphs. IV" to the list, but it doesn't matter because it's listed in (1998a).

Regarding sufficiently large gain groups on page 246, third paragraph (beginning "We shall asssume"): Stephen Gagola studied the order of A that is sufficient; see Zaslavsky (1998a), at author Gagola.

Sect. 15.2: The name should be "Unimodality conjectures". They are conjectures about unimodality, not conjectures that are unimodal. The literature says "unimodality", fortunately.