Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

Question

I found one rather interesting but intractable topology problem.

Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

Despite various attempts to contract the graph or reduce it to irreducible triangulation, nothing came of it.

What can you recommend?

Answer 1

This isn't a proof, but I'll post some references. The problem is a research-level one for sure, and looks a bit too involved for a post here.

The proof that such a triangulation is impossible is the content of the first part of the main proof in On the toroidal analogue of Eberhard's Theorem, by Jendrol and Jucovič, Proc. London Math. Soc. (3) 25 (1972), 385–398 (Thank-you to Lee Mosher for finding a full citation).

Effectively, the main theorem shows that the degree sequences of the form 5, 6, ... 6, 7 are the only degree sequences that 'work' with the Euler Characteristic which are not degree sequences of toroidal polyhedra.

If you prefer a different flavor proof via holonomy (?), you may be interested in There is no triangulation of the torus with vertex degrees 5, 6, . . . , 6, 7 and related results: Geometric proofs for combinatorial theorems, by Izmestiev, Kusner, Rote, Springborn and Sullivan.