>>2981681Not related to the question.
What you need to do is consider that a map M with no "trivalent vertex" in the map means that if you look at G(M), the graph associated to the map M, there are no 3 cycles.
Thus the girth of G(M) is at least 4, which allows you to re-prove the usual |E| >= 3|V| - 6 instead as |E| >= 2|V| - 4.
From here, I think it's clear you want to just apply induction. I won't do any details because you that would make you lazy.