Monochromatic partitions in 2-edge-coloured bipartite graphs

We study two variations of the Gyarfas--Lehel conjecture on the minimum number of monochromatic components needed to cover an edge-coloured complete bipartite graph. Specifically, we show the following. - For p>> (n/n) ^1/2, w. h. p. ~every 2-colouring of the random bipartite graph G~ G (n, n, p) admits a cover of all but O (1/p) vertices of G using at most three vertex-disjoint monochromatic components. - For every 2-colouring of a bipartite graph G with parts of size n and minimum degree (13/16+o (1) ) n, the vertices of G can be covered using at most three vertex-disjoint monochromatic components.