# Fast and Simple Edge-Coloring Algorithms

## TL;DR

This paper presents improved sequential algorithms for edge-coloring graphs efficiently with fewer colors, including a faster deterministic method and a simpler randomized approach, advancing the state-of-the-art in edge-coloring.

## Contribution

The authors develop a faster $O(|E|
oot sqrt{|V|})$ algorithm for $(d+1)$-edge-coloring and introduce a simple, high-probability randomized variant, along with a new algorithm for multigraphs.

## Key findings

- Improved runtime from $O(|E|
oot sqrt{|V|} ext{ to } O(|E|
oot sqrt{|V|})$ for edge-coloring.
- A simple randomized algorithm with same asymptotic runtime and high probability success.
- New algorithm for $(2d-1)$-edge-coloring of multigraphs with $O(|E| ext{log } d)$ time.

## Abstract

We develop sequential algorithms for constructing edge-colorings of graphs and multigraphs efficiently and using few colors. Our primary focus is edge-coloring arbitrary simple graphs using $d+1$ colors, where $d$ is the largest vertex degree in the graph. Vizing's Theorem states that every simple graph can be edge-colored using $d+1$ colors. Although some graphs can be edge-colored using only $d$ colors, it is NP-hard to recognize graphs of this type [Holyer, 1981]. So using $d+1$ colors is a natural goal. Efficient techniques for $(d+1)$-edge-coloring were developed by Gabow, Nishizeki, Kariv, Leven, and Terada in 1985, and independently by Arjomandi in 1982, leading to algorithms that run in $O(|E| \sqrt{|V| \log |V|})$ time. They have remained the fastest known algorithms for this task. We improve the runtime to $O(|E| \sqrt{|V|})$ with a small modification and careful analysis. We then develop a randomized version of the algorithm that is much simpler to implement and has the same asymptotic runtime, with very high probability. On the way to these results, we give a simple algorithm for $(2d-1)$-edge-coloring of multigraphs that runs in $O(|E|\log d)$ time. Underlying these algorithms is a general edge-coloring strategy which may lend itself to further applications.