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A244951 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).

%I A244951 #55 Feb 09 2018 03:30:40
%S A244951 4,3,2,4,3
%N A244951 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 A244951 Felix Fröhlich, <a href="/A244951/a244951_1.png">Illustration of colorings via Schlegel diagrams</a>
%H A244951 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 A244951 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 A244951 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 A244951 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 A244951 a(4) and a(5) are shown in illustration in the links.
%p A244951 with(GraphTheory): with(SpecialGraphs):
%p A244951 map(ChromaticNumber @ PlaneDual, [TetrahedronGraph(), HypercubeGraph(3), OctahedronGraph(), DodecahedronGraph(), IcosahedronGraph()]); # _Robert Israel_, Aug 24 2014
%Y A244951 Cf. A098112, A198861, A158478 (analog for sides of polygons).
%K A244951 nonn,fini,full
%O A244951 1,1
%A A244951 _Felix Fröhlich_, Jul 08 2014
%E A244951 Corrected value of a(4) due to discovery of a new coloring for the dodecahedron.
%E A244951 Corrected value of a(5) due to discovery of a new coloring for the icosahedron.