Switching 3-edge-colorings of cubic graphs

TL;DR

This paper investigates edge-Kempe switching in 3-edge-colorings of cubic graphs, providing computational data, new graph families, and connections to Steiner triple systems, advancing understanding of edge-coloring transformations.

Contribution

It introduces new families of cubic graphs with specific edge-Kempe properties, offers computational results up to certain graph sizes, and links edge-Kempe switching to cycle switching in Steiner triple systems.

Findings

Families of cubic graphs with many edge-Kempe classes are constructed.

Computational results for graphs up to order 30 (nonbipartite) and 36 (bipartite).

Conjecture that no cubic graphs have more than a certain number of edge-Kempe classes.

Abstract

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders $4n+2$ and $4n+4$ with $2^n$ edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equivalence class are also obtained. Edge-Kempe switching is further connected to cycle switching of Steiner triple systems, for which an improvement of the established classification algorithm is presented.