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On a recolouring version of Hadwiger's conjecture

Marthe Bonamy, Marc Heinrich, Clément Legrand-Duchesne, Jonathan Narboni

TL;DR

We address whether all $t$-colourings of graphs with no $K_t$-minor lie in a single Kempe equivalence class, and the related quasi-$K_t$-minor question. The authors build a random graph $G_n$ with a natural quasi-$K_n$-minor that, with high probability, has no $K_{(\tfrac{2}{3}+\varepsilon)n}$-minor. They also show the existence of a frozen $(\tfrac{3}{2}-\varepsilon)n$-colouring, along with a second, distinct colour partition, yielding a separation in Kempe reconfiguration and refuting the conjectures. The results thus separate quasi-minor structure from minors and reveal limitations of reconfiguration methods in Hadwiger-type settings, with implications for graph minor theory and reconfiguration complexity.

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.

On a recolouring version of Hadwiger's conjecture

TL;DR

We address whether all -colourings of graphs with no -minor lie in a single Kempe equivalence class, and the related quasi--minor question. The authors build a random graph with a natural quasi--minor that, with high probability, has no -minor. They also show the existence of a frozen -colouring, along with a second, distinct colour partition, yielding a separation in Kempe reconfiguration and refuting the conjectures. The results thus separate quasi-minor structure from minors and reveal limitations of reconfiguration methods in Hadwiger-type settings, with implications for graph minor theory and reconfiguration complexity.

Abstract

We prove that for any , for any large enough , there is a graph that admits no -minor but admits a -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.

Paper Structure

This paper contains 5 sections, 2 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Construction
  3. No large simple minor
  4. No large double minor
  5. No large triple minor

Key Result

Theorem 1.3

For every $\varepsilon>0$ and for any large enough $t$, there is a graph with no $K_t$-minor, whose $(\frac{3}{2}-\varepsilon)t$-colourings are not all Kempe equivalent.

Figures (1)

  • Figure 1: A different $n$-colouring given an appropriate quadruple.

Theorems & Definitions (11)

  • Conjecture 1.1: Conjecture A in las1981kempe
  • Conjecture 1.2: Conjecture A' in las1981kempe
  • Theorem 1.3
  • Conjecture 1.4: Conjecture C in las1981kempe
  • Theorem 1.5
  • Claim 2.1
  • Claim 2.2
  • Claim 2.3
  • proof : Proof of Claim \ref{['lem:simple']}
  • proof : Proof of Claim \ref{['lem:double']}
  • ...and 1 more