A309442 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).

5, 4, 2, 3, 5, 3

Offset: 1

Author

Sangeet Paul, Aug 03 2019

Keywords

• nonn
• fini
• full

Comments

Here, cells are 3-dimensional polyhedra, and faces are 2-dimensional polygons.

The sequence is the 4-dimensional analog of A244951.

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).

Examples

			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).
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).
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).
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.
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.
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.
		

Crossrefs

Cf. A244951, A273509.