Kempe equivalence of almost bipartite graphs

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of Kempe changes which in-terchange the colors used on a component of the subgraph induced by two color classes. It is PSPACE-complete to determine whether two given vertex k-colorings of a graph are Kempe equivalent for any fixed k & GE; 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 almost bipartite graphs which can be obtained from a bipartite graph by adding several edges, each connecting 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 was recently proved by Cranston and Feghali that this conjecture is false in general. This is a short version of the original paper [arXiv:2207.13244].