Weak maps and the Tutte Polynomial

TL;DR

This paper investigates how the Tutte polynomial behaves under weak maps between matroids, establishing conditions under which the polynomial's value decreases and exploring related implications.

Contribution

It generalizes Lucas's results by characterizing the Tutte polynomial's monotonicity under rank-preserving weak maps between matroids.

Findings

Tutte polynomial decreases under certain weak maps when x+y≥xy

Equality holds only when the matroids are isomorphic

Provides new insights into matroid invariants and their relations

Abstract

Let $M$ and $N$ be matroids such that $N$ is the image of $M$ under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for $x$ and $y$ positive, $T(M;x,y)\geq T(N;x,y)$ if and only if $x+y\geq xy$ or $M\cong N$. We give a number of consequences of this result.