# The NL-flow polynomial

## TL;DR

This paper introduces the NL-flow polynomial as an algebraic flow theory for digraphs, extending Tutte's nowhere-zero flow concepts to the dichromatic number and regular oriented matroids, with computational insights.

## Contribution

It defines NL-flows, derives their polynomial formula, and generalizes the flow equivalence theorem to oriented matroids, advancing algebraic flow theory for digraphs.

## Key findings

- Derived a closed formula for NL-flow polynomial.
- Extended flow theory to regular oriented matroids.
- Provided computational methods for complete digraph orientations.

## Abstract

In 1982 V\'{i}ctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow theory for the dichromatic number transferring Tutte's theory of nowhere-zero flows (NZ-flows) from classic graph colorings has been developed by Hochst\"attler. The purpose of this paper is to pursue this analogy by introducing a new definition of algebraic Neumann-Lara-flows (NL-flows) and a closed formula for their polynomial. Furthermore we generalize the Equivalence Theorem for nowhere-zero flows to NL-flows in the setting of regular oriented matroids. Finally we discuss computational aspects of computing the NL-flow polynomial for orientations of complete digraphs and obtain a closed formula in the acyclic case.