  We derive a general expression for standard tensor norm of $N$-body
correlation tensors for $N$-qubit complete graph states. With the help of this
expression, we formulate a separability criterion that identifies
non-$k$-separability of a class of $N$-qubit complete graph states, including
GHZ state. We illustrate the performance of our criterion by considering the
$N$-qubit complete graph states added with colored product noise. We also
demonstrate that few local measurement settings are sufficient to evaluate our
criterion.
