Single node of degree 5

We fail to prove this configuration colorable.

(A planar graph)

We color the boundary nodes with all 10 possible color combinations, modulo equivalence of colors.

We see no way to color the configuration given any of the 5 following boundary rings (though other rings, listed below, work):

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)


In case you were interested in the successful parts,

5 successes are simple and 0 require Kempe-chains.

Easily colorable boundary node color-arrangements

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)

(A planar graph)