# A conditional greedy algorithm for edge-coloring

## TL;DR

This paper introduces a new greedy algorithm for edge-coloring multigraphs, aiming to address a long-standing conjecture by proving correctness for certain cases.

## Contribution

The paper proposes a novel conditional greedy algorithm for edge-coloring multigraphs, potentially resolving a major open conjecture in the field.

## Key findings

- Algorithm correctness for multigraphs with chromatic index > max degree + 1
- Potential proof of a long-standing conjecture in edge-coloring
- Advances understanding of edge-coloring complexity

## Abstract

We present a novel algorithm for edge-coloring of multigraphs. The correctness of this algorithm for multigraphs with $\chi' > \Delta +1$ ($\chi'$ is the chromatic edge number and $\Delta$ is the maximum vertex degree) would prove a long standing conjecture in edge-coloring of multigraphs.