Fast Recoloring of Sparse Graphs

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 et al [8] 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 with a linear number of recolorings.