# Orientations of 1-Factorizations and the List Chromatic Index of Small Graphs

## TL;DR

This paper introduces a new corollary to the Quantitative Combinatorial Nullstellensatz that simplifies calculations related to 1-factorizations and applies it to investigate the List Edge Coloring Conjecture for small graphs.

## Contribution

It formulates a polynomial evaluation-based corollary, characterizes the sign of 1-factors, and uses these insights to computationally verify the conjecture for graphs up to 10 vertices.

## Key findings

- Corollary simplifies polynomial evaluations for combinatorial problems.
- Sign of 1-factorizations can be computed efficiently.
- Verification of the List Edge Coloring Conjecture for graphs with up to 10 vertices.

## Abstract

As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain situations, our corollary is more directly applicable and more ready-to-go than the Combinatorial Nullstellensatz itself. It is also of interest from a numerical point of view. We use it to explain a well-known connection between the sign of 1-factorizations (edge colorings) and the List Edge Coloring Conjecture. For efficient calculations and a better understanding of the sign, we then introduce and characterize the sign of single 1-factors. We show that the product over all signs of all the 1-factors in a 1-factorization is the sign of that 1-factorization. Using this result in an algorithm, we attempt to prove the List Edge Coloring Conjecture for all graphs with up to 10 vertices. This leaves us with some exceptional cases that need to be attacked with other methods.