  Non-contextuality (NC) and Bell inequalities can be expressed as bounds
$\Omega$ for positive linear combinations $S$ of probabilities of events, $S
\leq \Omega$. Exclusive events in $S$ can be represented as adjacent vertices
of a graph called the exclusivity graph of $S$. In the case that events
correspond to the outcomes of quantum projective measurements, quantum
probabilities are intimately related to the Gr\"otschel-Lov\'asz-Schrijver
theta body of the exclusivity graph. Then, one can easily compute an upper
bound to the maximum quantum violation of any NC or Bell inequality by
optimizing $S$ over the theta body and calculating the Lov\'asz number of the
corresponding exclusivity graph. In some cases, this upper bound is tight and
gives the exact maximum quantum violation. However, in general, this is not the
case. The reason is that the exclusivity graph does not distinguish among the
different ways exclusivity can occur in Bell-inequality (and similar)
scenarios. An interesting question is whether there is a graph-theoretical
concept which accounts for this problem. Here we show that, for any given
$N$-partite Bell inequality, an edge-coloured multigraph composed of $N$
single-colour graphs can be used to encode the relationships of exclusivity
between each party's parts of the events. Then, the maximum quantum violation
of the Bell inequality is exactly given by a refinement of the Lov\'asz number
that applies to these edge-coloured multigraphs. We show how to calculate upper
bounds for this number using a hierarchy of semi-definite programs and
calculate upper bounds for $I_3$, $I_{3322}$ and the three bipartite Bell
inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical
approach introduced here may remove some obstacles in the program of explaining
quantum correlations from first principles.
