On coupling of probability distributions and estimating the divergence through variation

Let $X$ be a discrete random variable with a given probability distribution. For any $\alpha$, $0\le\alpha\le1$, we obtain precise values for both the maximum and minimum variational distance between $X$ and another random variable $Y$ under which an $\alpha$-coupling of these random variables is possible. We also give the maximum and minimum values for couplings of $X$ and $Y$ provided that the variational distance between these random variables is fixed. As a consequence, we obtain a new lower bound on the divergence through variational distance.