Recoloring bounded treewidth graphs

Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Any graph is (tw+2)-mixing, where tw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two (tw+2)-colorings is at most quadratic, a problem left open in Bonamy et al. (2012).Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We improve Jerrumʼs bound using the grundy number, which is the worst number of colors in a greedy coloring.