Problem
Research questions and friction points this paper is trying to address.
Disproving conjectures on graph recolouring
Exploring frozen colourings in Kempe changes
Investigating Hadwiger's conjecture in graph theory
On a recolouring version of Hadwiger's conjecture
🤖 AI Summary
This paper investigates a recoloring variant of the Hadwiger conjecture, focusing on three conjectures by Las Vergnas and Meyniel (1981) concerning Kempe reconfigurability. The authors construct a family of $K_t$-minor-free graphs that, for all sufficiently large $t$ and any $varepsilon > 0$, admit a $(frac{3}{2} - varepsilon)t$-coloring in which the union of any two color classes induces a connected subgraph—rendering the coloring globally “frozen” under Kempe moves. This construction serves as the first counterexample to all three conjectures, exposing a fundamental incompatibility between minor-exclusion constraints and Kempe dynamics. Technically, the proof integrates extremal graph construction, refined Kempe chain analysis, and minor-exclusion arguments. The result establishes a new paradigm bridging graph recoloring theory and Hadwiger-type structural problems, advancing our understanding of the interplay between graph minors and reconfiguration landscapes.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Graph recolouring with frozen Kempe changes
Disproof of Las Vergnas and Meyniel conjectures
Use of large t and epsilon in graph theory
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