[Paper Review] Towards Cereceda's conjecture for planar graphs

Eduard Eiben, Carl Feghali|arXiv (Cornell University)|Oct 1, 2018

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper makes progress on Cereceda's conjecture by proving that the reconfiguration graph of 7-colourings for planar graphs has diameter at most $2^{\mathcal{O}(\sqrt{n})}$, significantly improving the prior bound of $\mathcal{O}(k^n)$. The approach uses structural decomposition of planar graphs into independent sets and 2-degenerate subgraphs, combined with recolouring sequences that reduce colour count efficiently via bounded-degree subgraphs and component analysis.

ABSTRACT

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on the colour of exactly one vertex. Cereceda conjectured ten years ago that, for every $k$-degenerate graph $G$ on $n$ vertices, $R_{k+2}(G)$ has diameter $\mathcal{O}({n^2})$. The conjecture is wide open, with a best known bound of $\mathcal{O}({k^n})$, even for planar graphs. We improve this bound for planar graphs to $2^{\mathcal{O}({\sqrt{n}})}$. Our proof can be transformed into an algorithm that runs in $2^{\mathcal{O}({\sqrt{n}})}$ time.

Motivation & Objective

• To address Cereceda's conjecture on the diameter of reconfiguration graphs for $k$-colourings of $d$-degenerate graphs, particularly for planar graphs.
• To close the gap between the conjectured quadratic diameter and the best-known $\mathcal{O}(k^n)$ bound, especially for $k=7$ in planar graphs.
• To establish a sub-exponential diameter bound for $R_7(G)$ in planar graphs $G$ with $n$ vertices.
• To provide a constructive algorithmic framework for transforming one 7-colouring into another using $2^{\mathcal{O}(\sqrt{n})}$ recolourings.

Proposed method

• Decompose the planar graph $G$ into a partition $V = I_1 \cup I_2 \cup A$, where $I_1$ and $I_2$ are independent sets and $A$ induces a 2-degenerate subgraph $H$.
• Use a recursive recolouring strategy to reduce from 7-colourings to 5-colourings via intermediate 6-colourings, leveraging bounded recolouring costs per vertex.
• Apply a component-based analysis using Jordan Curve Theorem arguments to ensure that recolouring sequences can be constructed without violating colour constraints.
• Utilize known results on degenerate graphs: for $k \geq 2d+1$, $R_k(G)$ has diameter $\mathcal{O}(n^2)$ when $G$ is $d$-degenerate.
• Apply a recurrence relation $f(n) \leq 5 \cdot f(n - n/49) + \mathcal{O}(n)$ to bound the number of recolourings per vertex, solved via the master theorem to yield $2^{\mathcal{O}(\sqrt{n})}$.
• Construct recolouring sequences that avoid colours used on adjacent vertices by analyzing alternating component structures in the colouring graph.

Experimental results

Research questions

• RQ1 Can the diameter of the reconfiguration graph $R_7(G)$ for planar graphs be bounded sub-exponentially?
• RQ2 Is it possible to reduce a 7-colouring of a planar graph to a 6-colouring using only polynomially many recolourings?
• RQ3 Does a partition of a planar graph into two independent sets and a 2-degenerate subgraph enable efficient recolouring sequences?
• RQ4 Can the structural decomposition of planar graphs be used to achieve sub-exponential diameter bounds in reconfiguration graphs?

Key findings

• The reconfiguration graph $R_7(G)$ for any planar graph $G$ on $n$ vertices has diameter at most $2^{\mathcal{O}(\sqrt{n})}$.
• The bound is achieved by reducing 7-colourings to 5-colourings via intermediate 6-colourings using $2^{\mathcal{O}(\sqrt{n})}$ recolourings.
• The method constructs a recolouring sequence from any 7-colouring to any other 7-colouring in sub-exponential time, with the algorithm running in $2^{\mathcal{O}(\sqrt{n})}$ time.
• The proof relies on a decomposition of planar graphs into two independent sets and a 2-degenerate subgraph, enabling controlled recolouring within the degenerate part.
• The recurrence $f(n) \leq 5 \cdot f(n - n/49) + \mathcal{O}(n)$ yields a sub-exponential bound via the master theorem, confirming the overall complexity.
• The result improves the prior best-known bound of $\mathcal{O}(k^n)$ for planar graphs, particularly for $k=7$, and provides a constructive algorithm.