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Pré-Publication, Document De Travail Année : 2022

Short and local transformations between ($\Delta+1$)-colorings

Résumé

Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring $\sigma$ to a target coloring $\eta$. Each pair of consecutive colorings must differ on exactly one vertex. The question becomes: is there a sequence of colorings from $\sigma$ to $\eta$? In this paper, we focus on $(\Delta+1)$-colorings of graphs of maximum degree $\Delta$. 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 $f(\Delta)$ independent sets one after another. Moreover, these independent sets can be computed efficiently in the LOCAL model of distributed computing.
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Dates et versions

hal-03663638 , version 1 (10-05-2022)

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Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, Mikaël Rabie. Short and local transformations between ($\Delta+1$)-colorings. 2022. ⟨hal-03663638⟩
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