Almost generalized uniform matroids and excluded minors

TL;DR

This paper characterizes almost $(k,l)$-uniform matroids using Tutte polynomial coefficients, identifies their excluded minors, and constructs an infinite family of such minors extending known classes.

Contribution

It provides a Tutte polynomial-based characterization of almost $(k,l)$-uniform matroids and describes their excluded minors, including an infinite family extending previous results.

Findings

Characterization of almost $(k,l)$-uniform matroids via Tutte polynomial coefficients.

Identification of excluded minors for these matroids.

Construction of an infinite family of excluded minors.

Abstract

We establish that matroids characterized by the Tutte polynomial $\sum_{i,j\ge 0}t_{i,j}x^iy^j$ with coefficients $t_{i,j}$ vanishing for $(i,j)\ge (k,l)$ precisely coincide with $(k,l)$-uniform matroids. This characterization implies that almost $(k,l)$-uniform matroids are exactly matroids with $t_{k,l}\le 1$ and $t_{i,j}=0$ if $(i,j)>(k,l)$. We also characterize excluded minors of almost $(k,l)$-uniform matroids in terms of Tutte polynomial coefficients. Finally, we construct an infinite family of excluded minors of almost $(k,l)$-uniform matroids which extend previously known cases of almost uniform and almost paving matroids.