Suppose I am given a spherical embedding of a graph, specifically a triangulation, onto the unit sphere such that all edges are short geodesics (lengths strictly smaller than $\pi$). Let us further call a vertex low degree if it has degree at most 5.
What is the largest number of low degree vertices $v$ that can exist in such an embedding such that $\text{star}(v)$ is not a subset of any open hemisphere?
Note that it is fine if the stars of different vertices overlap. Given the above, I am a little unsure how to approach an upper bound on this quantity but I have constructed an example that gives a lower bound of 3, as seen below:
Vertices 0 and 1 are vertices of degree 4 such that their stars do not fit in a hemisphere and vertex 2 is a degree 5 vertex whose star achieves the same. To further explain, vertices 1, 2, and 4 live on one great circle, vertices 0, 2, 4 lives on another great circle, and vertices 0, 1, 3, and 5 live on another great circle, where all three of these great circles are based along orthogonal directions.
Any advice on how to think about this? Any references on geometry that might be useful? I am relatively new to thinking about questions like this about spherical geometry.