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In terms of the $f$-vector for a translationally invariant (in $\Bbb R^3$) honeycomb define $$ \begin{split} v &= \max\left( \frac{f_1}{f_0}, \frac{f_2}{f_3}\right), \\ f &= \max\left( \frac{f_{12}}{f_1}, \frac{f_{12}}{f_2} \right), \end{split} $$ Taking a partial order on these pairs of numbers $(v, f)$ where $(v,f) \le (v’,f’)$ if $v\le v’$ and $f\le f’$,

  1. What is known about pairs $(v,f)$ that are minimal?

  2. In particular, are the pairs $(6,4)$, $(4,6)$, $(3,10)$ minimal?

Related question:

  1. Over self dual honeycombs, does the cubic lattice minimize the quantity $$g = \frac{f_{12} }{ f_0 }=\frac{ f_{12} }{ f_3 }\:\,?$$

The pairs in 2) are the basis of the quantum error correcting codes introduced by Nickerson and Bombin: https://arxiv.org/abs/1810.09621

[Background/motivation: The graphs in question are used to map the structure of quantum entanglement created between photons while building a photonic quantum computer. The reason to consider both the primal graph and its dual is to do with needing to protect complementary (non-commuting) quantum observables from noise. The reason to seek “low valence” examples is to do with the ease of creating the necessary entanglement. The reason to not care about convexity is to do with the fact that entanglement has a neat “malleability” property where it is unaffected by the absolute spatial location of the physical objects which are entangled. And finally, although the question is phrased in terms of 3d space, answers for higher dimensions are possibly also useful because photons can in principle be used to create entanglement corresponding to arbitrary graphs - although it takes more hardware to do so and so would only be practical if the noise tolerance went up dramatically.]

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