  We characterize the 1-unconditional subsequences of the canonical basis
(e_rc) of elementary matrices in the Schatten-von-Neumann class S^p . The set I
of couples (r,c) must be the set of edges of a bipartite graph without cycles
of even length 4<=l<=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 S^p_I spanned by
(e_rc)_{(r,c)\in I} in S^p .
