Coloured and Dependent Planar Matchings of Random Bipartite Graphs

TL;DR

This paper investigates rainbow and dependent planar matchings in random bipartite graphs, showing that large rainbow matchings exist with high probability and providing estimates for maximum planar matchings in dependent settings.

Contribution

It introduces new probabilistic bounds for rainbow and dependent planar matchings in bipartite graphs, extending understanding of their structure and size.

Findings

Large rainbow matchings occur with high probability when colors grow linearly with n.

Estimates for maximum planar matchings in dependent models are provided.

Implications for longest increasing subsequences in random permutations are discussed.

Abstract

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.