Rainbow $k$-connectivity of random bipartite graphs

TL;DR

This paper extends the understanding of rainbow $k$-connectivity, a measure of edge-coloring complexity ensuring multiple disjoint rainbow paths between vertices, from random graphs to bipartite graphs.

Contribution

It generalizes existing results on rainbow $k$-connectivity thresholds from $G(n,p)$ to bipartite graphs $G(m,n,p)$, providing new theoretical insights.

Findings

Established sharp threshold functions for rainbow $k$-connectivity in bipartite graphs.

Extended previous results from non-bipartite to bipartite random graphs.

Provided new bounds and probabilistic thresholds for edge-coloring properties.

Abstract

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 $G(m,n,p)$.