Toward Cereceda's conjecture for planar graphs

Abstract

The reconfiguration graph R k ( G ) of the k‐colorings of a graph G has as vertex set the set of all possible k‐colorings of G and two colorings are adjacent if they differ on the color of exactly one vertex. Cereceda conjectured 10 years ago that, for every k‐degenerate graph G on n vertices, R k + 2 ( G ) has diameter scriptO ( n 2 ). The conjecture is wide open, with a best known bound of scriptO ( k n ), even for planar graphs. We improve this bound for planar graphs to 2 scriptO ( n ). Our proof can be transformed into an algorithm that runs in 2 scriptO ( n ) time.

Decreasing the Maximum Average Degree by Deleting an Independent Set or a $d$-Degenerate Subgraph

“…As we have already mentioned, our results imply the positive answer to the last remaining step in the outline of the proof by Eiben and Feghali [16] of the following fact: Theorem 8. Let k 2 and l k + 2 be integers and let G be a graph on n vertices such that mad(G…”

“…Our results imply a positive answer for the open problem presented in [16] (Problem 2 from the final remarks), which implies a subexponential bound on the diameter of reconfiguration graphs of (k + 2)-colourings for graphs G with maximum average degree strictly less than k + 1. However, this bound has already been improved in [18] to a polynomial bound depending on the value of mad(G) in a slightly less general setting.…”

An Update on Reconfiguring $10$-Colorings of Planar Graphs

“…This bound would be best possible [3]. Although the conjecture has resisted several efforts, there have been some partial results surrounding it [1,6,5,9,10,11,13]. The most important breakthrough is a theorem of Bousquet and Heinrich [5] where it was shown, among other results, that R k+2 (G) has diameter O(n k+1 ).…”

Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

“…The result of Bousquet and Perarnau [13] relating the maximum average degree of a graph and the recolouring diameter of a graph (see Section 3.2) implies that the 8recolouring diameter of a planar graph is polynomial in n. This bound was improved by Feghali [31] who showed that the 8-recolouring diameter of a planar graph is O(n(log n) 7 ). Eiben and Feghali [28] showed that the 7-recolouring diameter of a planar graph is at most 2 O( √ n) . Feghali [32] showed that the 10-recolouring diameter of a planar graph is at most n 2 .…”