Edge Kempe equivalence of regular graph covers

TL;DR

This paper proves that for any finite d-regular graph with a proper edge coloring, all other colorings can be reached through finitely many edge Kempe switches on an appropriate finite cover, with the cover size depending only on d.

Contribution

It establishes that all proper edge colorings of a regular graph are Kempe equivalent on some finite cover, extending Kempe equivalence concepts to graph covers.

Findings

Any proper edge coloring can be transformed into any other via Kempe switches on a suitable cover.

The covering degree needed depends only on the regular degree d of the graph.

The result applies to finite regular graphs with proper edge colorings.

Abstract

Let $G$ be a finite $d$-regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of $G$ obtained by switching the two colors along some bi-chromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that $G$ is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on $d$.