Positroids are 3-colorable

TL;DR

This paper proves that all orientations of positroids of rank at least 3 are 3-colorable, establishing a significant property of their chromatic number related to oriented matroids.

Contribution

It demonstrates that every orientation of a positroid of rank at least 3 has a good coline, implying 3-colorability of all such oriented matroids.

Findings

Every positroid of rank ≥ 3 has a good coline.

All orientations of these positroids are 3-colorable.

The result links positroid structure to oriented matroid colorability.

Abstract

We show that every positroid of rank $r \geq 3$ has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J.\ Ne\v{s}et\v{r}il, R.\ Nickel, and W.~Hochst\"{a}ttler, this shows that every orientation of a positroid is 3-colorable.