Recolouring weakly chordal graphs and the complement of trianglefree graphs
Abstract
For a graph $G$, the $k$recolouring graph $\mathcal{R}_k(G)$ is the graph whose vertices are the $k$colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. We prove that for all $n \ge 1$, there exists a $k$colourable weakly chordal graph $G$ where $\mathcal{R}_{k+n}(G)$ is disconnected, answering an open question of Feghali and Fiala. We also show that for every $k$colourable $3K_1$free graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4V(G)$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.11087
 arXiv:
 arXiv:2106.11087
 Bibcode:
 2021arXiv210611087M
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics
 EPrint:
 6 pages, 2 figures