# Show that each edge of the cyclic polytope $C_4(6)$ is contained in either three or four facets, and either three or four 2-faces.

Note: here $$C_4(6)$$ is the notation for the cyclic polytope of dimension 4 and of 6 vertices.

By the 2-neighbourly property of $$C_4(6)$$ and the Dehn-Sommerville equations, I've determined that the polytope has 6 vertices, 15 edges, 18 2-faces and 9 3-faces. Also, since cyclic polytopes are simplicial, their k-faces have exactly (k+1) vertices.

However, these are not enough to answer the questions above. What other characterization of cyclic polytopes that could help me with those questions?

Since the dimension 4 is even, each vertex figure of $$C_4(6)$$ is combinatorially equivalent to $$C_3(5)$$, a triangular bipyramid. (You can find a proof of this statement in Facets and Nonfacets of Convex Polytopes by Perles and Shephard, statement (6), on page 117.)
So, considering the vertex figure at a vertex $$v$$ of $$C_4(6)$$, there are two vertices of valence three (corresponding to edges incident to $$v$$ contained in three 2-faces and three facets), and three vertices of valence 4 (corresponding to edges incident to $$v$$ contained in four 2-faces and four facets).
Moreover, we can count that there are $$6 \cdot 2 / 2 = 6$$ edges of the first type, and $$6 \cdot 3 / 2 = 9$$ edges of the second type.
• Can you please elaborate on your first claim that the vertex figure of $C_4(6)$ is equivalent to $C_3(5)$?