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).

4, 3, 2, 4, 3

Offset

1,1

Links

Table of n, a(n) for n=1..5.

Felix Fröhlich, Illustration of colorings via Schlegel diagrams

Martin Gardner, The Five Platonic Solids, in Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions, Cambridge University Press, (see page 6).

Example

a(1) = 4, since in the tetrahedron any face shares a common edge with any other face, so each face needs a distinct color.
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.
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.
a(4) and a(5) are shown in illustration in the links.

Maple

with(GraphTheory): with(SpecialGraphs):
map(ChromaticNumber @ PlaneDual, [TetrahedronGraph(), HypercubeGraph(3), OctahedronGraph(), DodecahedronGraph(), IcosahedronGraph()]); # Robert Israel, Aug 24 2014

Crossrefs

Cf. A098112, A198861, A158478 (analog for sides of polygons).

Sequence in context: A184338 A184412 A304240 * A110631 A333669 A159846

Adjacent sequences: A244948 A244949 A244950 * A244952 A244953 A244954

Keyword

nonn,fini,full

Author

Felix Fröhlich, Jul 08 2014

Extensions

Corrected value of a(4) due to discovery of a new coloring for the dodecahedron.

Corrected value of a(5) due to discovery of a new coloring for the icosahedron.

Status

approved