We characterise the 1-unconditional subsets $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices in the Schatten–von-Neumann class $\mathrm{S}^p$. The set of couples $I$ must be the set of edges of a bipartite graph without cycles of even length $4 \lel \le p$ if $p$ is an even integer, and without cycles at all if $p$ is a positive real number that is not an even integer. In the latter case, $I$ is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space $\mathrm{S}^p_I$ spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.