Transversal matroids and the half plane property

Abstract

We focus on checking the validity of the half-plane property on two prominent classes of transversal matroids, namely lattice path matroids and bicircular matroids. We show that lattice path matroids satisfy the half-plane property. Subsequently, we show an explicit example of a bicircular matroid that is not a positroid and discuss the negative correlation properties of bases of transversal matroids. We prove that sparse paving matroids do not satisfy the Rayleigh property, which helps us gain new perspectives about conjectures on negative correlation in basis elements of matroids in general.

Figures (6)

• Figure 1: $\theta-$graph, loose handcuff and tight handcuff graphs.
• Figure 2: An example of a lattice path matroid (a) $\mathsf{M} = \mathsf{M}[12378,678910]$ and (b) represents an example of a snake.
• Figure 3: The matroid $\mathsf{M}$ discussed in Example \ref{['eg:Example_principal']} along with its truncation and principal extension.
• Figure 4: A rank four bicircular matroid $M$ which is not a positroid with one of its bicircular representations.
• Figure 5: A Venn diagram depicting how the classes of transversal matroids, positroids, bicircular matroids, lattice path matroids, multi path matroids, lattice path bicircular matroids intersect in the superclass of gammoids.

Theorems & Definitions (36)

• Theorem 1
• Corollary 2
• Definition 3
• Definition 4: Definition 3.1 bonin2003lattice
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Definition 10