Recoloring graphs of treewidth 2

12/21/2020

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by   Valentin Bartier, et al.

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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.