Reconfiguring 10-colourings of planar graphs

02/06/2019

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by   Carl Feghali, et al.

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Let k ≥ 1 be an integer. 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 exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer ℓ≥ k + 2 and k-degenerate graph G on n vertices, R_ℓ(G) has diameter O(n^2). The conjecture has been verified only when ℓ≥ 2k + 1. We give a simple proof that if G is a planar graph on n vertices, then R_10(G) has diameter at most n^2. Since planar graphs are 5-degenerate, this affirms Cereceda's conjecture for planar graphs in the case ℓ = 2k.