Seven-color requirement when unit-circle arcs and higher-degree trichromatic vertices are allowed

Establish whether 7 colors are necessary to color the plane with a locally finite map-type coloring when region boundaries may contain arcs of unit circles and trichromatic vertices of degree greater than 3 are permitted.

Background

The main theorem establishes a 7-color lower bound only after imposing two structural restrictions: boundaries cannot contain arcs of unit curvature, and trichromatic vertices of degree greater than 3 are forbidden. The second concluding question asks whether the same lower bound remains valid for all locally finite map-type colorings after both restrictions are removed.