Field Types

A trivial matroid is the matroid whose only independent set is the empty set. Its rank is .
A discrete matroid on a non-empty finite set is a matroid in which every subset of is an independent set
A -uniform matroid is a matroid on a non-empty finite set , whose bases are those subsets with exactly -elements. If has elements, then we denote this matroid as .
- The independent of a uniform matroid are the subsets of with no more than elements.
- It also follows that the rank is defined as
A representable matroid is a matroid over a set over a field where there exists a vector space over and a map such that a subset is independent in if and only if is linearly independent.
- In other words, we may map all independent sets of the matroid to linearly independent subsets of the vector space.
- A matroid is representable if such a field exists.
- Representable matroids form an isomorphism class
A regular matroid is representable over every field.
A binary matroid is a matroid that is representable over the field of integers modulo .
The Fano Matroid is the matroid defined on the set whose bases are all the subsets of , except , , , , , and .
- The Fano Matroid is binary and Eulerian
- The Fano Matroid is not graphic, cographic, transversal, nor regular

The cycle matroid also called the graphic matroid of a graph is an isomorphism class of matroids associated with graph denoted

It is defined as the matroid on the set of edges of , with the bases being the spanning forests of .
- In a graphic matroid, a loop generalizes the graph theory loops and parallel elements generalize multiple edges.
- Isomorphism preserves cycles, cut-sets and the number of edges, but not necessarily connectedness the number of vertices or their degrees
The cutset matroid also called the cographic matroid of a graph is an isomorphism class of matroids associated with denoted

It is the matroid on the set of edges of with the bases being the cotrees and the cycles being the cut-sets
A Planar Matroid is a matroid that is both graphic and cographic.

It corresponds to planar graphs.
A bipartite matroid is a matroid in which each cycle has an even number of elements.
An Eulerian Matroid is a matroid on if can be written as a set of disjoint of disjoint cycles.

Let be a non-empty finite set and a family of subsets of . Then, the transversal matroid is a matroid where the independent sets are the partial transversals of

We denote this matroid as .
Up to isomorphism, the number of transversal matroids where is
Every transversal matroid is representable over some field .
Every transversal matroid is binary if and only if it is graphic.

Other Theorems

(Wilson e31.6) Every uniform matroid is transversal.
- Proof: If is a -uniform matroid, then , where consists of copies of .
Tutte’s Theorems
- A binary matroid is regular if and only if it contains no minor isomorphic to the Fano Matroid or its dual
- (Wilson 32.9) A matroid is graphic if and only if it is binary and it contains no minor isomorphic to or to the Fano Matroid and its dual.
- (Wilson 32.9) A matroid is cographic if and only if it is binary and it contains no minor isomorphic to or to the Fano Matroid and its dual.
Kuratowski’s Theorem Analogue — A matroid is planar if and only if it regular, and contains no minor isomorphic to or their duals. ²