An update on reconfiguring 10-colorings of planar graphs

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.