An update on reconfiguring 10-colorings of planar graphs (Q2223443)

Summary: 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 \textit{N. Bousquet} and \textit{G. Perarnau} [Eur. J. Comb. 52, Part A, 1--11 (2016; Zbl 1327.05189)] 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.