Fast Recoloring of Sparse Graphs

Abstract: In this paper, we show that for every graph of maximum average degree bounded away from $d$, any $(d+1)$-coloring can be transformed into any other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any $8$-coloring of a planar graph into any other $8$-coloring with a polynomial number of recolorings. These results give some evidence on a conjecture of Cereceda, van den Heuvel and Johnson which asserts that any $(d+2)$ coloring of a $d$-degenerate graph can be transformed into any other one using a polynomial number of recolorings. We also show that any $(2d+2)$-coloring of a $d$-degenerate graph can be transformed into any other one using a linear number of recolorings.