On a recolouring version of Hadwiger's conjecture

TL;DR

The paper constructs graphs that challenge existing conjectures by showing they can be colored with fewer colors than their minors would suggest, while maintaining a 'frozen' coloring structure.

Contribution

It provides a counterexample to longstanding conjectures by demonstrating graphs with no large clique minors yet with specific constrained colorings.

Findings

Existence of graphs with no large $K_t$-minor but with a $(rac{3}{2}- ext{small } ext{epsilon})t$-coloring.

Such colorings are 'frozen', meaning each color class induces a connected subgraph.

Disproves three conjectures from 1981 regarding Hadwiger's conjecture and graph coloring.

Abstract

We prove that for any $\varepsilon>0$, for any large enough $t$, there is a graph $G$ that admits no $K_t$-minor but admits a $(\frac32-\varepsilon)t$-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.