Mediated Digraphs and Quantum Nonlocality

Abstract

A digraph D=(V,A) is mediated if, for each pair x,y of distinct vertices of D, either xy belongs to A or yx belongs to A or there is a vertex z such that both xz,yz belong to A. For a digraph D, DELTA(D) is the maximum in-degree of a vertex in D. The "nth 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)=<f(n)+c. Methods and results of graph theory, design theory and number theory are used.