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Towards Cereceda's conjecture for planar graphs
- Publication Year :
- 2018
-
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.<br />Comment: 12 pages
- Subjects :
- Computer Science - Discrete Mathematics
Mathematics - Combinatorics
05C15
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1810.00731
- Document Type :
- Working Paper