LECTURES ON THE THEORY OF MATROIDS JACK E. GRAVER Abstract. These are the lecture notes for a short course pre- sented at the University of Alberta in Edmonton, Alberta, in March of 1966. The first two thirds of these notes give an introduction to the theory of matroids and is based on two fundamental papers of the subject: [2], On the abstract properties of linear dependence; and [1], Lectures on Matroids. In the last third of these notes, matroid theory is applied to the theory of graphs. Most of the results obtained here were originally worked out by H. Whitney in a series of papers preceding [2]. In fact it is evident that the work that went into these papers led to and finally culminated in his matroid papers. Consider a finite dimensional vector space V over a field K. It is clear how we define the collection I of independent sets. Furthermore it is easy to prove: (i 1 ) a subset of an independent set is independent (i 2 ) if I and J are independent sets such that |I | < |J |, then there exists a vector v 2 J \ I where I [ {v} is independent. Likewise we can define the collection B of bases for V and prove: (b 1 ) all bases have the same number of vectors; (b 2 ) if B 1 and B 2 are bases, v 2 B 1 , then there exists a w 2 B 2 so that (B 1 \{v}) [ {w} is a basis. Also we can define the rank function r on subsets of V (the dimension of the subspace spanned by the vectors in that subspace) and prove: (r 0 ) the rank of the empty set is zero; (r 1 ) r(S [ {v} is equal to r(S ) or r(S ) + 1; (r 2 ) if r(S [ {v})= r(S [ {w})= r(S ), then r(S [ {v,w})= r(S ). Finally we can define the collection D of minimal dependent sets and prove: (d 1 ) If D is a minimal dependent set and E is a proper subset of D, then E is not a minimal dependent set; (d 2 ) If D 1 and D 2 are minimal dependent sets, v 2 D 1 \D 2 , w 2 D 1 \ D 2 then there exists a minimal dependent set D 3 so that w 2 D 3 ✓ ((D 1 [ D 2 ) \{v}). Date : September 1966. 1