The Merino--Welsh conjecture for split matroids

TL;DR

This paper proves the Merino--Welsh conjecture for a broad class of split matroids by leveraging their lattice of cyclic flats, extending the conjecture's validity beyond previously known cases.

Contribution

It establishes the conjecture for split matroids, a large class including paving and copaving matroids, using their lattice structure.

Findings

The conjecture holds for all split matroids.

Split matroids include paving and copaving matroids.

The proof uses the structure of cyclic flats.

Abstract

In 1999 Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.