%I #55 Feb 09 2018 03:30:40
%S 4,3,2,4,3
%N Minimum number of colors needed to color the faces of the Platonic solids such that no two faces meeting at a common edge share the same color (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
%H Felix Fröhlich, <a href="/A244951/a244951_1.png">Illustration of colorings via Schlegel diagrams</a>
%H Martin Gardner, <a href="http://assets.cambridge.org/97805217/56105/excerpt/9780521756105_excerpt.pdf">The Five Platonic Solids</a>, in Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions, Cambridge University Press, (see page 6).
%e a(1) = 4, since in the tetrahedron any face shares a common edge with any other face, so each face needs a distinct color.
%e a(2) = 3, since the cube has three sets of opposite faces. Any two faces that are not opposite share a common edge, so only opposite faces can have the same color.
%e a(3) = 2, since cutting the octahedron along its "equator" results in two square pyramids. The triangular faces of a single pyramid can be colored using two colors in an alternating fashion. Then the two pyramids are reassembled such that at the "equator" differently colored faces meet.
%e a(4) and a(5) are shown in illustration in the links.
%p with(GraphTheory): with(SpecialGraphs):
%p map(ChromaticNumber @ PlaneDual, [TetrahedronGraph(), HypercubeGraph(3), OctahedronGraph(), DodecahedronGraph(), IcosahedronGraph()]); # _Robert Israel_, Aug 24 2014
%Y Cf. A098112, A198861, A158478 (analog for sides of polygons).
%K nonn,fini,full
%O 1,1
%A _Felix Fröhlich_, Jul 08 2014
%E Corrected value of a(4) due to discovery of a new coloring for the dodecahedron.
%E Corrected value of a(5) due to discovery of a new coloring for the icosahedron.