A path in an edge-colored graph $G$ is called a rainbow path if no two edges of the path are colored the same. The minimum number of colors required to color the edges of $G$ such that every pair of vertices are connected by at least $k$ internally vertex-disjoint rainbow paths is called the rainbow $k$-connectivity of the graph $G$, denoted by $rc_k(G)$. For the random graph $G(n,p)$, He and Liang got a sharp threshold function for the property $rc_k(G(n,p))\leq d$. In this paper, we extend this result to the case of random bipartite graph $G(m,n,p)$.

中文翻译：

---

随机二部图的彩虹 k 连通性

如果路径的两条边的颜色不同，则边着色图 $G$ 中的路径称为彩虹路径。为 $G$ 的边着色所需的最小颜色数，使得每对顶点至少通过 $k$ 内部顶点不相交的彩虹路径连接，称为图 $G$ 的彩虹 $k$-connectivity，由 $rc_k(G)$ 表示。对于随机图 $G(n,p)$，He 和 Liang 得到了属性 $rc_k(G(n,p))\leq d$ 的尖锐阈值函数。在本文中，我们将此结果扩展到随机二部图 $G(m,n,p)$ 的情况。