Quantum motivation
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[Noncontextuality inequalities](http://arxiv.org/abs/1102.0264) (and in particular Bell inequalities) can be [mapped into graphs](http://arxiv.org/abs/1010.2163), in such a way that its relevant properties can be calculated via some simple graph-theoretical functions. In particular, the maximal quantum violation of a noncontextuality inequality is upperbounded by the [Lovász function](http://en.wikipedia.org/wiki/Lovasz_number) of its graph.


A family of inequalities that interests myself particularly is represented by the [Möbius ladder](http://en.wikipedia.org/wiki/M%C3%B6bius_ladder); I was unable to find its Lovász function in the literature, but from [physical and mathematical grounds](http://arxiv.org/abs/1206.3212) we managed to prove that
$$\frac{n}{2}\bigg(1+\cos\frac{\pi}{n}\bigg) \le\vartheta(M\_{2n}) \le \frac{n\big(2\cos(\pi/n)+1\big)}{2+\cos(\pi/n)}.$$
Note that both bounds coincide in the asymptotic limit. Furthermore, my physical intuition claims that the function is always equal to this lower bound. I cannot prove it, however.


Quantum-free question
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Let $M\_{2n}$ be the [Möbius ladder](http://en.wikipedia.org/wiki/M%C3%B6bius_ladder) graph. I conjecture ts [Lovász function](http://en.wikipedia.org/wiki/Lovasz_number) to be


$$\vartheta(M\_{2n}) = \frac{n}{2}\bigg(1+\cos\frac{\pi}{n}\bigg).$$
But I can not prove this. I hope that this problem would be easy to a graph-theoretician, or even buried in the literature somewhere. Anybody knows?



Lovász function of the Möbius ladder
Now, 9 years later, this conjecture has been proven by [Bharti et al.](https://arxiv.org/abs/2104.13035) (one of the authors is Adán Cabello, who was in the paper that originally posed the conjecture).


The technique used was semidefinite programming (SDP) duality: computing the Lovász function is an SDP, and as all SDPs it has a dual SDP that upperbounds the solution of the primal SDP. They managed to find a feasible solution of the dual for all even $n$ that coincides with the already known lower bound that came from a feasible solution of the primal, proving that both solutions are optimal and give the Lovász function.


