Recoloring graphs of treewidth 2

TL;DR

This paper proves that for graphs of treewidth 2, a linear transformation exists between 5-colorings, advancing understanding of recoloring sequences in low-treewidth graphs.

Contribution

It establishes the existence of a linear recoloring sequence between 5-colorings in graphs of treewidth 2, reducing previous bounds for this class.

Findings

Linear transformation exists between 5-colorings for treewidth 2 graphs.

No linear transformation exists between 4-colorings for some treewidth 2 graphs.

Completes the characterization of recoloring bounds for treewidth 2 graphs.

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 note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist.