2-着色随机二部图中的精确三组件覆盖
Exact three-component covers in 2-coloured random bipartite graphs
- Universidade Federal de Pernambuco（伯南布哥联邦大学）
- Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决了随机二部图中二色三组件猜想，证明当边概率远大于根号下对数n除以n时，任何红蓝边着色都能用至多三个单色连通组件覆盖所有顶点，方法基于公共邻域扩展引理。
AI中文摘要：
我们解决了Fernández、Pavez-Signé和Stein关于随机二部图的二色三组件猜想。更精确地，我们证明如果$G\sim G(n,n,p)$且$p\gg\sqrt{\log n/n}$，则高概率下$G$的每个红蓝边着色都允许其顶点集被至多三个单色连通组件覆盖。证明基于公共邻域并集的均匀扩展引理和交替公共邻域扩展论证。
英文摘要：
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.
