Planar graphs have exponentially many 3-arboricities

TL;DR

This paper proves that planar graphs have exponentially many 3-colorings where each color class forms a forest, extending to 3-list-colorings, highlighting the abundance of such colorings.

Contribution

It establishes the exponential number of 3-colorings with forest color classes for all planar graphs, including the list-coloring variant, which was previously unknown.

Findings

Exponential lower bound on the number of 3-colorings with forest classes

Extension of results to 3-list-colorings

Confirms sharpness of the 3-coloring bound for planar graphs

Abstract

It is well-known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any (projective) planar graph. The same result holds in the setting of 3-list-colorings.