Short and local transformations between (Δ+1)-colorings
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Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring σ to a target coloring η. Each pair of consecutive colorings must differ on exactly one vertex. The question becomes: is there a sequence of colorings from σ to η? In this paper, we focus on (Δ+1)-colorings of graphs of maximum degree Δ. Feghali, Johnson and Paulusma proved that, if both colorings are non-frozen (i.e. we can change the color of a least one vertex), then a quadratic recoloring sequence always exists. We improve their result by proving that there actually exists a linear transformation. In addition, we prove that the core of our algorithm can be performed locally. Informally, if we start from a coloring where there is a set of well-spread non-frozen vertices, then we can reach any other such coloring by recoloring only f(Δ) independent sets one after another. Moreover, these independent sets can be computed efficiently in the LOCAL model of distributed computing.
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