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%I #11 Oct 27 2019 11:24:06
%S 5,4,2,3,5,3
%N Minimum number of colors needed to color the cells of the six regular convex polychora such that no two cells with a common face share the same color (in the order 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell).
%C Here, cells are 3-dimensional polyhedra, and faces are 2-dimensional polygons.
%C The sequence is the 4-dimensional analog of A244951.
%C The sequence is also the minimum number of colors needed to color the vertices of the six regular convex polychora such that no two vertices with a common edge share the same color (in the order 5-cell, 16-cell, 8-cell, 24-cell, 600-cell, 120-cell).
%e a(1) = 5, since in the 5-cell, each cell has a common face with every other cell (analogous to the tetrahedron, where each face has a common edge with every other face).
%e a(2) = 4, since in the 8-cell, each cell has a common face with every other cell except its "opposite" cell (analogous to the cube, where each face has a common edge with every other face except its opposite face).
%e a(3) = 2, since the 16-cell's dual graph has no odd-edge cycles (analogous to the octahedron's dual graph having no odd-edge cycles).
%e a(4) = 3, since the 24-cell has at least one 3-color solution, and its dual graph has a 3-vertex subgraph with no 2-color solution.
%e a(5) = 5, since the 120-cell has at least one 5-color solution, and its dual graph has a 30-vertex subgraph with no 4-color solution.
%e a(6) = 3, since the 600-cell has at least one 3-color solution, and its dual graph has a 5-vertex subgraph with no 2-color solution.
%Y Cf. A244951, A273509.
%K nonn,fini,full
%O 1,1
%A _Sangeet Paul_, Aug 03 2019
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