On a recolouring version of Hadwiger's conjecture

Abstract

We prove that for any $\epsilon >0$, for any large enough t, there is a graph that admits no $K_t$-minor but admits a $(\frac{3}{2}-\epsilon )t$-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.