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I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track down a proof or paper on the subject.

Is this actually true?

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  • $\begingroup$ What are the def's and details? $\endgroup$
    – Wlod AA
    Commented Jun 1, 2023 at 5:48
  • $\begingroup$ What do you mean by "cycle graph"? In the terminology I'm used to, these are just the graphs $C_n$. But then the result is trivial, isn't it? $\endgroup$
    – Emil Jeřábek
    Commented Jun 1, 2023 at 7:13
  • $\begingroup$ @EmilJeřábek I don't fully see it yet tbh, since to me it seems one needs to rescale there unit sphere... $\endgroup$
    – Justin_other_PhD
    Commented Jun 5, 2023 at 19:54
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    $\begingroup$ But any cyclic graph with $n$ vertices is isometric to a circumference of radius $n/(2\pi)$, right? So in that case it should be trivial (in fact spheres of radius $n/(2\pi)$ are the only ones which will allow the imbedding). In what I say I am considering cycle graphs as CW-complexes of dimension $1$ with edges of length $1$ $\endgroup$
    – Saúl RM
    Commented Jun 5, 2023 at 20:29
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    $\begingroup$ I see. Even considering the discrete version, for $n\geq4$ the cycle $C_n$ only imbeds in spheres of radius $n/(2\pi)$: this is because for any two points $p,q,r$ of the sphere, $d(p,q)+d(p,r)=d(q,r)$ implies that $p,q,r$ are in some geodesic, due to, for example, the spherical cosine theorem $\endgroup$
    – Saúl RM
    Commented Jun 5, 2023 at 23:13

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