A Characterization of Uniform Matroids

This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.


Introduction
Sets and their characteristic vectors will not be distinguished. We refer to Oxley 1 for the terminology about matroids and to Schrijver 2 for the terminology about polyhedra.
Let E be a finite set, and let M be a matroid defined on E. If M is 2-connected, then we will say that a proper subset L of E; that is, Ø / L / E, is locked if L is nonseparable or 2-connected in M, E \ L is nonseparable or 2-connected in M * and r L ≥ max{2, 2 r E − | E \ L|} or min{r L , r * E \ L } ≥ 2. Observe that L is locked in a matroid M if and only if M | L and M/L are both connected and min{r L , r * E \ L } ≥ 2. Locked subsets give the nontrivial facets of the bases polytope. We will denote the class of these subsets by . . , k, and the class of locked subsets Λ M is the union of such classes in each 2-connected component involved in the direct sums. Locked subsets were introduced by Chaourar 3-5 to describe some facets of the cone and the polytope generated by the matroid bases. We will denote the 2-sum of two matroids M and N using the basepoint {e} by M⊕ e N or M⊕ 2 N if there is no confusion. Since 2-sums with U 1,2 and U 1,1 are, respectively, identity and deletion, then we will consider only proper 2-sums without U 1,2 nor U 1,1 .
Let M be the class of matroids obtained by means of 1-sums or direct sums and proper 2-sums of uniform matroids together with all minors of such matroids. Since M ⊕ N * M * ⊕ N * and M⊕ 2 N * M * ⊕ 2 N * see 1 , then M is closed under the taking of duals. It is also clear that M is closed under the taking of minors.
If M is not a 3-connected matroid, then, using a theorem of Oxley see 1 , M can be constructed from 3-connected minors of it by a sequence of the operations of 1-sum and 2-sum.
The purpose of this paper is to characterize uniform matroids by means of locked subsets. There are exactly five 3-connected matroids of rank 3 on a 6-element set. These matroids can be obtained from M K 4 by relaxing zero, one, two, three, or four circuithyperplanes. The matroids are, respectively, M K 4 , the rank-3 whirl W 3 , Q 6 , P 6 , and the uniform matroid U 3,6 see 1 .
The remaining of the paper is organized as follows: in Section 2, we will give a characterization of uniform matroids by means of locked subsets, two consequences are given in Section 3, and the conclusion is given is Section 4.

The Characterization
We will need to three lemmas in this section.
Proof. Direct from the definition of a locked subset. The following last lemma of this section was proved by Walton 6 and we give here a new proof based on locked subsets. Proof. Suppose by contradiction that M is not uniform. It follows that there exists a subset F of M such that |F| r M and F contains a circuit C. Without loss of generality, we can suppose that |C| 3 because, if it is not, we can contract some elements of C keeping C as a circuit and decreasing its cardinality. Now we delete all elements of F − C. Let N be the obtained matroid. Here we give our main result.

Theorem 2.4. If M is a 3-connected matroid, then the following assertions are equivalent:
i M is a uniform matroid, Proof. i ⇒ ii Using the fact that there is a unique closed and 2-connected subset which is E.
ii ⇒ i Using Lemma 2.2, any minor N of M verifies Λ N Ø. So M has no isomorphic minor to any of M K 4 , W 3 , Q 6 , and P 6 , because any of these excluded minors has at least one locked subset circuit of rank 3 . By Lemma 2.3, M is uniform.
Note that i implies ii , in Theorem 2.4, even if M is not 3-connected.

Some Consequences
We will give two corollaries of our characterization.
The first one is a characterization by excluded minors and is almost a restatement of Lemma 2.3, and Walton should be credited for this result: Proof. i ⇒ ii By contradiction, suppose that M has one isomorphic to any of the excluded minors. Since all the excluded minors are 3-connected, then at least one of the 3-connected components used to construct M by means of 1-sums and 2-sums has one such excluded minor. Let N be this excluded minor. Since the number of locked subsets for any excluded minor is at least 1, then, using Lemmas 2.2 and 2.3 and Theorem 2.4, Λ N / Ø and N is not uniform.
ii ⇒ i If M is 3-connected, then, by Lemma 2.3, M is uniform. If M is not 3connected, then M can be construct using 3-connected matroids by means of 1-sum and 2-sum. It follows that no one of these matroids has an isomorphic to any of the excluded minors and, by Lemma 2.3, all these matroids are uniform.
We will need the following result of Chaourar

Conclusion
We have given a characterization of uniform matroids by means of locked subsets and two consequences of this characterization.