Binary Matroids Without K5 -minors
The complete graph on 5 vertices is denoted K5 . It’s corresponding matroid, denoted
M (K5 ) is binary and can be represented by the following matrix





1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
0
0
1
0
1
0
1
0
0
1
0
1
1
0
0
1
0
1
0
0
1
1





Let M be the class of (simple) binary matroids which do not contain M (K5 ) as a minor.
For r = 1, 2, 3, . . . , let h(r) denote the maximum size of a matroid in M or rank r. According
to Gordon Royle’s computations, the following information is known for r ≤ 8:
Table 1: Values for h(r)
r
1
2
3
4
5
6
7
8
h(r)
1
3
7
12
16
21
25
30
Number of Matroids
1
1
1
1
1
1
4
3
Based on the above numbers, Kung, Mayhew, Pivotto, and Royle (KMPR) conjectured
the following:
0.1 Conjecture
h(r) = 92 − a, where a = 13
2 if r is odd and a = 6 if r is even. Moreover, the extremal examples
are generalised parallel connections along a lone of copies of C12 and at most one copy of F7
(the Fano plane).
The question is, what are these “extremal examples” exactly? What is the matroid C12 ?
This is the matroid obtained from P G(3, 2) by deleting three collinear points. It’s represented
by the matrix below:
1

C12


=

1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
1
0
1
0
1
0
0
1
1
1
1
0
0
0
1
1
0
1
0
1
1
1
1
1
0
1
1
1
1





The obvious question to ask here is, why does this have no M (K5 )-minor? One way is
to look at the geometric representation of M (K5 ). There are 4 lines (ie. triangles) where no
point contains all lines. One can view P G(3, 2) as the matroid one obtains from P G(2, 2) (the
Fano plane) by adding a point, say x, and then joining x by lines to each point in the Fano
plane. The Fano plane has the property that any two lines intersect. Thus if we delete a line
from the Fano plane, we get a matroid with no (3-point) lines. Thus in P G(3, 2) if we delete a
line from the Fano plane, then every 3-point line must contain x. This means that C12 can’t
have a M (K5 )-minor because there is no such element in M (K5 ).
The next question is, why is C12 the unique matroid of rank 4 in M having a maximum
size? Is there a simple explanation for this? From the table, there is also a unique matroid
in M or rank 5 having maximum size. How do we construct this matroid? The answer is we
use a generalized parallel connection. The idea is, we take the matroids C12 and P G(2, 2) and
“glue” the matroids together along a line. The matroid we get has size 12 + 4 = 16 and it has
rank 4. In general, if one takes two matroids in M, say M1 and M2 , and glue them together
along a line, then one obtains a matroid M which also belongs to M. Why does this work?
The reason has do with the connectedness of matroids.
A matroid M is connected if for any two elements e, f of M , there is at least one circuit
containing e, f. In general, for a matroid M and subset X ⊆ E(M ), the connectivity of X is
defined to be
λ(X) = r(X) + r(E − X) − r(E).
A k-separation of M is a partition (X, E − X) of E(M ) where |X| > k, |E − X| > k and
λ(X) ≤ k −1. We define the connectivity of M to be λ(M ) which is the smallest value of k such
that M has a k-separation. Note that M (K5 ) has no k-separation for any k. If we go back to
our example where M is obtained by gluing M1 and M2 along a line, we see that λ(M ) ≤ 3.
Moreover, if M did have a M (K5 )-minor, then it would have to be contained in either M1 or
M2 (which can’t happen). So M does not contain a M (K5 )-minor.
2