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

Abstract

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph~$G$ has as vertex set the set of all possible proper $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. A result of Bousquet and Perarnau (2016) regarding graphs of bounded degeneracy implies that if $G$ is a planar graph with $n$ vertices, then $R_{12}(G)$ has diameter at most $6n$. We improve on the number of colors, showing that $R_{10}(G)$ has diameter at most $8n$ for every planar graph $G$ with $n$ vertices.

A Thomassen-type method for planar graph recoloring

“…Conjecture 22. Let G be a planar graph, let L be a list assignment for G, and let ϕ and ϕ ′ be L-colorings of G. If either As mentioned in [3], it is possible that the number 10 of colors in the statement of Theorem 3 is not the best possible. Perhaps a more sophisticated multiphase recoloring process might allow one to replace 10 by a smaller integer and still obtain a linear bound on the diameter of the reconfiguration graph.…”

“…Theorem 3 (Dvořák and Feghali [3]). If G is a planar graph on n vertices, then R 10 (G) has diameter at most 8n.…”

“…The proof in [3] is based on discharging and reducible configurations, and is quite long and involved. In this paper, we present a shorter proof based on a different idea.…”

Recolouring planar graphs of girth at least five

“…To the best of our knowledge, Conjecture 1.6 is only known to hold for outerplanar graphs [1] and 1-degenerate graphs. For partial results, Bartier and Bousquet [5] proved that G(G, d + 4) has diameter O(n) for every d-degenerate chordal graph G of bounded maximum degree, and Dvořák and Feghali proved that G(G, 10) has diameter O(n) for every planar graph G [11,12], and if G is triangle-free, then so does G(G, 7) [12].…”

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

“…Feghali [32] showed that the 10-recolouring diameter of a planar graph is at most n 2 . This was improved by Dvořák and Feghali [26] who showed that the 10-recolouring diameter of a planar graph is at most 8n. The results of Bousquet and Heinrich [12] imply that the 7-recolouring diameter of a planar graph is O(n 6 ) and for k ≥ 9, the k-recolouring diameter of a planar graph is O(n 2 ).…”