Recoloring graphs of treewidth 2
Valentin Bartier, Nicolas Bousquet, Marc Heinrich

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 -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 . Bousquet and Perarnau proved that a linear transformation exists for between -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.
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