Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

TL;DR

This paper investigates the number of edge-Kempe equivalence classes in 3-edge-colored cubic graphs, establishing unique class results for certain bipartite graphs and exploring infinite families with varying class counts.

Contribution

It provides new results on edge-Kempe equivalence classes in cubic graphs, including a unique class result for 2-connected planar bipartite graphs and constructions for nonplanar cases.

Findings

Every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class.

Infinite families of nonplanar bipartite cubic graphs with multiple classes are constructed.

Techniques developed can analyze other graph classes.

Abstract

Two edge colorings of a graph are {\em edge-Kempe equivalent} if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.