Recoloring graphs of treewidth 2

Abstract

Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any (d+2)-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for d=1. Bousquet and Perarnau proved that a linear transformation exists for between (2d+2)-colorings. It is open to determine if this bound can be reduced.In this paper, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. More formally, we prove that there always exists a linear transformation between any pair of 5-colorings. This result is tight since there exist graphs of treewidth 2 and two 4-colorings such that a shortest transformation between them is quadratic.