Open Problems

Abstract

The Grothendieck constant K(G) of a graph G = (V, E) is the least constant K such that for every A: E → R,

$$ \begin{gathered} \mathop {sup}\limits_{f:V \to S^{\left| V \right| - 1} } \sum\limits_{\{ u,v\} \in E} {A(u,v) \cdot \left\langle {f(u),f(v)} \right\rangle } \hfill \\ \leqslant K\mathop {sup}\limits_{f:V \to \{ - 1, + 1\} } \sum\limits_{\{ u,v\} \in E} {A(u,v) \cdot f(u)f(v)} . \hfill \\ \end{gathered} $$

This notion was introduced in [2], where it is shown that there is an absolute positive constant c so that for every graph G, K(G) ≤c log $\vartheta (\bar G)$, with $\vartheta (\bar G)$ being the Lovász theta function of the complement of G, defined in [12].