An infinite family of excluded minors for strong base-orderability

Abstract

We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the class of gammoids. We prove that a paving matroid is base-orderable if and only if it has no minor that is isomorphic to the cycle matroid of the complete graph on four vertices. For each k that is at least 2, we define the property of k-base-orderability, which lies strictly between base-orderability and strong base-orderability, and we show that k-base-orderable matroids form what Ingleton called a complete class. By generalizing an example of Ingleton, we construct a set of matroids, each of which is an excluded minor for k-base-orderability, but is (k-1)-base-orderable; the union of these sets, over all k, is an infinite set of base-orderable excluded minors for strong base-orderability.

Figures (6)

• Figure 1: The cycle matroid $M(K_4)$ and its basis-exchange digraph for the bases $A = \{a,b,c\}$ and $B=\{d,e,f\}$.
• Figure 2: The critical graph $\Delta_5$.
• Figure 3: The lattice of cyclic flats of $M(\Delta_5)$.
• Figure 4: The pair $(\{a_3, a_4\}, \{b_3, b_4\})$ is an obstruction in this critical graph, $\Delta_7$. The edges with gray arrows show that conditions (2) and (3) in Definition \ref{['def:obstruction']} hold.
• Figure 5: An example, with $r=8$, of the digraph $\Delta_\alpha$ described above. An arrow from block $U$ to block $V$ means that there is a directed edge $(u, v)$ for every $u \in U$ and $v \in V$.

Theorems & Definitions (71)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Definition 3.1