# Vizing's edge-recoloring conjecture holds

## TL;DR

This paper proves Vizing's edge-recoloring conjecture for all graphs, showing that starting from any k-edge-coloring, one can reach a Δ-edge-coloring using Kempe swaps, extending previous results limited to special graph classes.

## Contribution

The paper establishes the validity of Vizing's conjecture for all graphs, confirming that Δ-edge-colorings are reachable from any initial coloring via Kempe swaps.

## Key findings

- Vizing's conjecture is proven for all graphs.
- Kempe swaps suffice to reach Δ-edge-colorings.
- Extension of previous results from special graph classes.

## Abstract

In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($\Delta$ + 1)-edge-coloring of G where $\Delta$ is the maximum degree of G. One year later he conjectured that one can also reach a $\Delta$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.

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Source: https://tomesphere.com/paper/2302.12914