  For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a
graph, that is, assignments of $k$ distinct natural numbers to each node in
such a way that nodes that are connected by an edge receive numbers that are
strictly alternating between them with respect to the relation $<$. If it takes
at least $\chi_{int}^k(G)$ distinct numbers to provide graph $G$ with such a
coloring, then the interleaved multichromatic number of $G$ is
$\chi_{int}^*(G)=\inf_{k\ge 1}\chi_{int}^k(G)/k$ and is known to be given by a
function of the simple cycles of $G$ under acyclic orientations if $G$ is
connected [1]. This paper contains a new proof of this result. Unlike the
original proof, the new proof makes no assumptions on the connectedness of $G$,
nor does it resort to the possible applications of interleaved $k$-tuple
colorings and their properties.
