  The paper settles the problem of the consistency of the existence of a single
universal graph between a strong limit singular and its power. Assuming that in
a model of $\mathbf{GCH}$ $\kappa$ is supercompact and the cardinals $\theta <
\kappa$, $\lambda > \kappa$ are regular, as an application of a more general
method we obtain a forcing extension in which $\textrm{cf}(\kappa) = \theta$,
the Singular Cardinal Hypothesis fails at $\kappa$ and there exists a universal
graph in cardinality $\lambda \in (\kappa,2^\kappa)$.
