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

Abstract

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 (∆ + 1)-colorings of graphs of maximum degree ∆. 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 nonfrozen vertices, then we can reach any other such coloring by recoloring only f (∆) independent sets one after another. Moreover these independent sets can be computed efficiently in the LOCAL model of distributed computing.