  In this article we study the minimum number $\kappa$ of additional automata
that a Boolean automata network (BAN) associated with a given block-sequential
update schedule needs in order to simulate a given BAN with a parallel update
schedule. We introduce a graph that we call $\mathsf{NECC}$ graph built from
the BAN and the update schedule. We show the relation between $\kappa$ and the
chromatic number of the $\mathsf{NECC}$ graph. Thanks to this $\mathsf{NECC}$
graph, we bound $\kappa$ in the worst case between $n/2$ and $2n/3+2$ ($n$
being the size of the BAN simulated) and we conjecture that this number equals
$n/2$. We support this conjecture with two results: the clique number of a
$\mathsf{NECC}$ graph is always less than or equal to $n/2$ and, for the
subclass of bijective BANs, $\kappa$ is always less than or equal to $n/2+1$.
