Let \(k \ge 1\) 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 \(\ell \ge k + 2\) and k-degenerate graph G on n vertices, \(R_{\ell }(G)\) has diameter \(O(n^2)\). The conjecture has been verified only when \(\ell \ge 2k + 1\). We give a simple proof that if G is a planar graph on n vertices, then \(R_{10}(G)\) has diameter at most \(n(n + 1)/ 2\). Since planar graphs are 5-degenerate, this affirms Cereceda’s conjecture for planar graphs in the case \(\ell = 2k\).

令\（k \ ge 1 \）为整数。该重新配置图形\（R_k（G）\）的的ķ的曲线图的-colourings  ģ具有作为顶点设定的设定的所有可能的ķ的-colourings ģ如果它们不同于正好一个顶点和两个色素是相邻的。Cereceda于2007年的一个猜想断言，对于n个顶点上的每个整数\（\ ell \ ge k + 2 \）和k-退化图G，\（R _ {\ ell}（G）\）的直径为\（O（ n ^ 2）\）。仅当\（\ ell \ ge 2k +1 \）时才验证了猜想。我们举一个简单的证明，如果g ^是在n个顶点上的平面图，则\（R_ {10}（G）\）的直径最大为\（n（n + 1）/ 2 \）。由于平面图是5退化的，因此在\（\ ell = 2k \）的情况下肯定了Cereceda对平面图的猜想。