Stable chromatic number of planar graphs

Determine the stable chromatic number of planar graphs, including the asymptotically tight dependence on the number of vertices between the established lower bound of Ω(log n) and upper bound of O(log² n).

Background

The paper defines the stable chromatic number of a graph as the minimum number of colors guaranteed to suffice for a stable proper coloring under every preference profile. For planar graphs, it proves an O(log² n) upper bound using the planar product structure theorem and a reachability-based coloring method.

The paper also establishes an Ω(log n) lower bound for planar graphs because trees are planar and there exist n-vertex trees with stable chromatic number Ω(log n). Thus, the precise asymptotic behavior for planar graphs remains unresolved, with a gap of one logarithmic factor between the known bounds.