Kempe equivalence of almost bipartite graphs

TL;DR

This paper investigates the Kempe equivalence of graphs close to bipartite, proposing a conjecture, providing partial proofs, and discussing its eventual refutation by recent research.

Contribution

It introduces a conjecture on Kempe equivalence in almost bipartite graphs and offers partial solutions and insights into its validity.

Findings

Conjecture proposed for Kempe equivalence in almost bipartite graphs.

Partial solutions and bounds provided for the conjecture.

Recent work disproved the conjecture in general.

Abstract

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of switchings of two colors of vertices. It is PSPACE-complete to determine whether two given vertex $k$-colorings of a graph are Kempe equivalent for any fixed $k\geq 3$, and it is easy to see that every two vertex colorings of any bipartite graph are Kempe equivalent. In this paper, we consider Kempe equivalence of {\it almost} bipartite graphs which can be obtained from a bipartite graph by adding several edges to connect two vertices in the same partite set. We give a conjecture of Kempe equivalence of such graphs, and we prove several partial solutions and best possibility of the conjecture, but it is more lately proved by Cranston and Feghali that this conjecture is false in general.