Exact three-component covers in 2-coloured random bipartite graphs

Abstract

We resolve the two-colour three-component conjecture of Fern\'andez, Pavez-Sign\'e and Stein for random bipartite graphs. More precisely, we prove that if $G\sim G(n,n,p)$ and $p\gg\sqrt{\log n/n}$, then with high probability every red--blue edge-colouring of $G$ admits a cover of its vertex set by at most three monochromatic connected components. The proof is based on a uniform expansion lemma for unions of common neighbourhoods and an alternating common-neighbourhood expansion argument.