  Suppose $G$ and $H$ are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a
partition of $V(H)$ such that the subgraph of $H$ induced between $L(v)$ and
$L(v')$ is a matching whenever $vv'\in E(G)$. We show for each $\varepsilon>0$
that, if $H$ has maximum degree $D$ and $|L(v)| \ge (1+\varepsilon)D/\log D$
for all $v\in V(G)$, then $H$ admits an independent transversal with respect to
$L$, provided $D$ is sufficiently large. This bound on the part sizes is
asymptotically sharp up to a factor $2$. We also show some asymmetric variants
of this result.

Combinatorics