Mediated Digraphs and Quantum Nonlocality
Gregory Gutin, Nick Jones, Arash Rafiey, Simone Severini and Anders Yeo
Discrete Applied Mathematics
A digraph $D=(V,A)$ is mediated if for each pair $x,y$ of distinct
vertices of $D$, either $xy\in A$ or $yx\in A$ or there is a vertex
$z$ such that both $xz,yz\in A.$ For a digraph $D$, $\Delta^-(D)$ is
the maximum in-degree of a vertex in $D$. The $n$th mediation number
$\mu (n)$ is the minimum of $\Delta^-(D)$ over all mediated digraphs
on $n$ vertices. Mediated digraphs and $\mu(n)$ are of interest in
the study of quantum nonlocality.
We obtain a lower bound $f(n)$ for $\mu(n)$ and determine infinite
sequences of values of $n$ for which $\mu(n)=f(n)$ and
$\mu(n)>f(n)$, respectively. We derive upper bounds for $\mu(n)$ and
prove that $\mu(n)=f(n)(1+o(1))$. We conjecture that there is a
constant $c$ such that $\mu(n)\le f(n)+c.$ Methods and results of
design theory and number theory are used.