Year: 2026
Type: preprint
Abstract: We resolve the two-colour three-component conjecture of Fernández, Pavez-Signé 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.
Source: arXiv (Cornell University)
Authors: Xiao-Chuan Liu, X U Yang
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213432323