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Anton Petrunin
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First let me reformulate the problem in a more geometric way.

Let $\Gamma$ be the metric space glued from segments of length $\pi/n$ along the rule described by your graph.

Note that $\Gamma$ is a $CAT(1)$ space of diameter $\pi$ with extendable geodesics. It seems that any such $\Gamma$ is obtained by gluing few copies of $[0,\pi]$ along the ends.

Assume the later is proved. Take the images of the ends, say $x,y\in \mathbb S^n$. Let $z$ be the midpoint of the minimizing geodesic from $x$ and $y$ in $\mathbb S^n$. Then the image of each arc and therefore the image of whole $\Gamma$ lies in the half-sphere with center $z$.

Anton Petrunin
  • 45k
  • 14
  • 135
  • 299