Wilson, exercise 10.4

We fail to prove this configuration colorable.

(A planar graph)

We color the boundary nodes with all 27 possible color combinations, modulo equivalence of colors, that are consistent with this smaller configuration:

(A planar graph)

We see no way to color the configuration given any of the 8 following boundary rings (though other rings, listed below, work):

In case you were interested in the successful parts,

18 successes are simple and 1 requires Kempe-chains.

Easily colorable boundary node color-arrangements

Kempe chain possibilities for RB/GY on

with color-flipped boundary nodes marked with heavy dashes. (Apologies for the kempe chain edges passing through things visually.)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)