# Expansion and contraction functors on matriods

## TL;DR

This paper investigates how expansion and contraction functors affect matroid properties, showing that certain properties are preserved under expansions and introducing contraction as a new tool to analyze White's conjecture.

## Contribution

It introduces the contraction functor for matroids, establishes the equivalence of properties under expansions, and links White's conjecture to expansions and contractions.

## Key findings

- Expansion preserves graphic, binary, and transversal properties.
- White's conjecture holds for a matroid if and only if it holds for all its expansions.
- Some classes of matroids are identified that satisfy White's conjecture.

## Abstract

Let $M$ be a matroid. We study the expansions of $M$ mainly to see how the combinatorial properties of $M$ and its expansions are related to each other. It is shown that $M$ is a graphic, binary or a transversal matroid if and only if an arbitrary expansion of $M$ has the same property. Then we introduce a new functor, called contraction, which acts in contrast to expansion functor. As a main result of paper, we prove that a matroid $M$ satisfies White's conjecture if and only if an arbitrary expansion of $M$ does. It follows that it suffices to focus on the contraction of a given matroid for checking whether the matroid satisfies White's conjecture. Finally, some classes of matroids satisfying White's conjecture are presented.