A Note on Spanning Trees in Near \(d\)-angulations

A near \(d\)-angulation is a planar graph in which every region has degree \(d\) except for the boundary region. Let \(T\) be a spanning tree with all of its vertices of odd degree on the boundary. Then the interior regions can be 2-colored so that regions that share edges of \(T\) receive different colors and regions which share edges not in \(T\) receive the same color. The boundary region is given a third color. We prove that the number of regions of each color can be determined from only knowing the behavior on the boundary.