  Using one channel to simulate another exactly with the aid of quantum
no-signalling correlations has been studied recently. The one-shot
no-signalling assisted classical zero-error simulation cost of non-commutative
bipartite graphs has been formulated as semidefinite programms [Duan and
Winter, IEEE Trans. Inf. Theory 62, 891 (2016)]. Before our work, it was
unknown whether the one-shot (or asymptotic) no-signalling assisted zero-error
classical simulation cost for general non-commutative graphs is multiplicative
(resp. additive) or not. In this paper we address these issues and give a
general sufficient condition for the multiplicativity of the one-shot
simulation cost and the additivity of the asymptotic simulation cost of
non-commutative bipartite graphs, which include all known cases such as
extremal graphs and classical-quantum graphs. Applying this condition, we
exhibit a large class of so-called \emph{cheapest-full-rank graphs} whose
asymptotic zero-error simulation cost is given by the one-shot simulation cost.
Finally, we disprove the multiplicativity of one-shot simulation cost by
explicitly constructing a special class of qubit-qutrit non-commutative
bipartite graphs.
