Fast Recoloring of Sparse Graphs

TL;DR

This paper demonstrates efficient methods for recoloring sparse graphs with bounded average degree, showing polynomial and linear bounds for transforming colorings, thus supporting conjectures in graph recoloring theory.

Contribution

It introduces polynomial and linear bounds for recoloring sparse graphs, advancing understanding of graph recoloring transformations and supporting existing conjectures.

Findings

Polynomial number of recolorings for certain graph classes.

Linear number of recolorings for $d$-degenerate graphs.

Supports conjecture on recoloring bounds for 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.