  In this short note, we show that given a cost function $c$, any coupling
$\pi$ of two probability measures where the second is a discrete measure can be
associated to a certain bipartite graph containing a perfect matching, based on
the value of the infinity transport cost $\norm{c}_{L^\infty(\pi)}$. This
correspondence between couplings and bipartite graphs is explicitly
constructed. We give two applications of this result to the
$\mathcal{W}_\infty$ optimal transport problem when the target measure is
discrete, the first is a condition to ensure existence of an optimal plan
induced by a mapping, and the second is a numerical approach to approximating
optimal plans.
