# How many complete triangulations of the sphere are there in which every vertex has exactly degree five?

I know of one with 12 vertices, 20 faces, and 30 edges. Are there any others? Almost sorry I asked the question. The answer is trivial. Use Euler's formula and the unique solution pops out.

Let $v,e,f$ be the number of vertices, edges and faces. So by your given condition, we have $e = 5v/2 , f = 5v/3$. And use this in the euler characteristic formula for sphere $v-e+f =2$ to get $v (1-5/2+5/3) = 2 \implies v = 12$. So there are only one such triangulation.

• Is it possible that there would be two non-isomorphic triangulations with the same number of vertices, edges, and faces? No of course, but is there an easy way of seeing this? – Tyler Seacrest Apr 14 '15 at 15:26
• I think it is possible that two non-isomorphic graph has same v, e, f. But I do not know the details. – aNumosh Apr 14 '15 at 20:03