Boundary Values in the Four Color Problem

Let be a planar graph drawn in the plane so that its outer boundary is a -cycle. A four coloring of the outer boundary is admissible if there is a four coloring of which coincides with on the boundary. If is the number of admissible boundary colorings, we show that the 4CC implies <!-- MATH $\psi \geqslant 3 \cdot {2^k}$ --> for <!-- MATH $k = 3, \cdots ,6$ --> . We conjecture this to be true for all and show is <!-- MATH $\geqslant c{((1 + {5^{1/2}})/2)^k}$ --> .