List-recoloring of sparse graphs

Abstract

Fix a graph G, a list-assignment L for G, and L-colorings α and β. An L-recoloring sequence, starting from α, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper L-coloring. An L-recoloring sequence transforms α to β if its initial coloring is α and its final coloring is β. We prove there exists an L-recoloring sequence that transforms α to β and recolors each vertex at most a constant number of times if (i) G is triangle-free and planar and L is a 7-assignment, or (ii) mad(G)<17/5 and L is a 6-assignment or (iii) mad(G)<22/9 and L is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvořák and Feghali.