Strengthening a theorem of Meyniel

01/19/2022

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by   Quentin Deschamps, et al.

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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.