Kempe equivalence of edge-colourings in subcubic and subquartic graphs

TL;DR

The paper proves that all 4-edge-colourings of subcubic graphs are Kempe equivalent, resolving a conjecture, and identifies the maximum degree threshold for Kempe equivalence in edge-colourings.

Contribution

It establishes Kempe equivalence for 4-edge-colourings in subcubic graphs and extends the result to (Delta+2)-edge-colourings for Delta up to 4, identifying degree thresholds.

Findings

All 4-edge-colourings of subcubic graphs are Kempe equivalent.

Kempe equivalence does not hold for 4-edge-colourings when Delta=4.

For Delta ≤ 4, all (Delta+2)-edge-colourings are Kempe equivalent.

Abstract

It is proved that all 4-edge-colourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Delta=3 is a threshold for Kempe equivalence of (Delta+1)-edge-colourings, as such an equivalence does not hold in general when Delta=4. One extra colour allows a similar result in this latter case however, namely, when Delta<=4 it is shown that all (Delta+2)-edge-colourings are Kempe equivalent.