Kempe classes and almost bipartite graphs

Let G be a graph and k be a positive integer, and let Kc(G,k) denote the number of Kempe equivalence classes for the k-colorings of G. In 2006, Mohar noted that Kc(G,k)=1 if G is bipartite. As a generalization, we show that Kc(G,k)=1 if G is formed from a bipartite graph by adding any number of edges less than ⌈k/2⌉2+⌊k/2⌋2. We show that our result is tight (up to lower order terms) by constructing, for each k≥8, a graph G formed from a bipartite graph by adding (k2+8k−45+1)/4 edges such that Kc(G,k)≥2. This refutes a recent conjecture of Higashitani–Matsumoto.