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In this paper, we study two problems related to planar matchings in randombipartite 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-trivialfraction of ${r}$ with high probability, i.e. with probability converging toone as ${n \rightarrow \infty}$ Next we consider planar matchings in adependent setting where each vertex is forced to choose exactly one neighbourfrom amongst all possible choices. We obtain estimates for the largest size ofa planar matching and also discuss the implication of our results to longestincreasing subsequences in enlarged random permutations.

Coloured and Dependent Planar Matchings of Random Bipartite Graphs