Publication at Faculty of Mathematics and Physics |
2020
Abstract
Let π>=1 be an integer. The reconfiguration graph π π(πΊ) 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 exactly one vertex.
A conjecture of Cereceda from 2007 asserts that for every integer β>=π+2 and k-degenerate graph G on n vertices, π β(πΊ) has diameter π(π2). The conjecture has been verified only when β>=2π+1.
We give a simple proof that if G is a planar graph on n vertices, then π 10(πΊ) has diameter at most π(π+1)/2. Since planar graphs are 5-degenerate, this affirms Cereceda's conjecture for planar graphs in the case β=2οΏ½οΏ½.
