Publication at Faculty of Mathematics and Physics |
2020
Abstract
Let ๐>=1 be an integer. The reconfiguration graph ๐ ๐(๐บ) 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 โ>=๐+2 and k-degenerate graph G on n vertices, ๐ โ(๐บ) has diameter ๐(๐2). The conjecture has been verified only when โ>=2๐+1.
We give a simple proof that if G is a planar graph on n vertices, then ๐ 10(๐บ) has diameter at most ๐(๐+1)/2. Since planar graphs are 5-degenerate, this affirms Cereceda's conjecture for planar graphs in the case โ=2๏ฟฝ๏ฟฝ.