  In this paper, we study two problems related to planar matchings in random
bipartite graphs. First, we colour each edge of the complete bipartite graph
$K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$
grows linearly with ${n}$ then the maximum rainbow matching is a non-trivial
fraction of ${r}$ with high probability, i.e. with probability converging to
one as ${n \rightarrow \infty}$ Next we consider planar matchings in a
dependent setting where each vertex is forced to choose exactly one neighbour
from amongst all possible choices. We obtain estimates for the largest size of
a planar matching and also discuss the implication of our results to longest
increasing subsequences in enlarged random permutations.
