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For an integer k ≥ 1 and a graph G, let K k (G) be the graph that has vertex set all proper k-colorings of G, and an edge between two vertices α and β whenever the coloring β can be obtained from α by a single Kempe change. A theorem of Meyniel from 1978 states that K 5 (G) is connected with diameter O(5 |V (G)| ) for every planar graph G. We significantly strengthen this result, by showing that there is a positive constant c such that K 5 (G) has diameter O(|V (G)| c ) for every planar graph G.

Let G be a connected graph with maximum degree ∆ ≥ 3 distinct from K∆+1. Generalizing Brooks' Theorem, Borodin, Kostochka and Toft proved that if p1, . . . , ps are non-negative integers such that p1 +• • • +ps ≥ ∆−s, then G admits a vertex partition into parts A1, . . . , As such that, for 1 ≤ i ≤ s, G[Ai] is pi-degenerate. Here we show that such a partition can be performed in linear time. This generalizes previous results that treated subcases of a conjecture of Abu-Khzam, Feghali and Heggernes [2], which our result settles in full.