Polyhedral graph such that every vertex has degree $2k$, for some $k > 2$

Question

Is there any polyhedral graph such that every vertex has degree even greater than 4?

Answer 1

If a graph's every vertex has degree of at least $6$, then it cannot be planar graph because it needs to satisfy Euler's formula.

See this MO question's comment by Noam Elkies for details.

Answer 2

Let $V$, $F$, $E$ be the number of vertices, faces and edges. If every vertex has degree $\ge 6$, $E \ge 3 V$. Every face has at least three edges, so $E \ge 3 F/2$. Thus the Euler characteristic $V-E+F < E/3 - E + 2 E/3 = 0$. This can't occur for a spherical polyhedron or a handle-body, though I suppose it could for some exotic non-orientable "polyhedron".