Old Bulgaria problem

[Румен Симеонов]

Old Bulgaria problem, accordind to MelonGirl:
https://artofproblemsolving.com/communi ... 4p13514379
MelonGirl
#1VPMNov 20, 2019, 6:32 AM• 1 Y
A triangulation of a regular $n$-gon is fully even if the number of diagonals coming out of each vertex is even. Find all possible $n,$ for which there exists a fully even triangulation of a regular $n$-gon.

[Румен Симеонов]

Are srar and/or skew regular n-gons under cosideration? I woud assume as obvios fact the assertion that such ones don't have triangulations and, consequently, it remains to solve the problem only amongst convex regular n-gons:
https://en.m.wikipedia.org/wiki/Regular_polygon
What triangulation means in this problem? Are admissible triangulatios with additional vertexes, ar only triangulatios withe diagonals are assumed?