On Rainbow-$k$-Connectivity of Random Graphs

A path in an edge-colored graph is called a \emph{rainbow path} if all edges on it have pairwise distinct colors. For $k\geq 1$, the \emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the minimum number of colors required to color the edges of $G$ in such a way that every two distinct vertices are connected by at least $k$ internally disjoint rainbow paths. In this paper, we study rainbow-$k$-connectivity in the setting of random graphs. We show that for every fixed integer $d\geq 2$ and every $k\leq O(\log n)$, $p=\frac{(\log n)^{1/d}}{n^{(d-1)/d}}$ is a sharp threshold function for the property $rc_k(G(n,p))\leq d$. This substantially generalizes a result due to Caro et al., stating that $p=\sqrt{\frac{\log n}{n}}$ is a sharp threshold function for the property $rc_1(G(n,p))\leq 2$. As a by-product, we obtain a polynomial-time algorithm that makes $G(n,p)$ rainbow-$k$-connected using at most one more than the optimal number of colors with probability $1-o(1)$, for all $k\leq O(\log n)$ and $p=n^{-\epsilon(1\pm o(1))}$ for some constant $\epsilon\in[0,1)$.

💡 Research Summary

The paper investigates the rainbow‑k‑connectivity rcₖ(G) of the Erdős–Rényi random graph G(n,p). A rainbow path is a path whose edges receive pairwise distinct colors, and a graph is rainbow‑k‑connected if every pair of distinct vertices is linked by at least k internally disjoint rainbow paths. The authors extend the earlier result of Caro, Lev, et al., which identified a sharp threshold p = √(log n / n) for the property rc₁(G) ≤ 2, to the much broader setting of any fixed integer d ≥ 2 and any k up to O(log n).

The main theorem states that for every fixed d ≥ 2 and every k = O(log n), the function

 p* = (log n)^{1/d} / n^{(d‑1)/d}

is a sharp threshold for the event rcₖ(G(n,p)) ≤ d. In other words, if p = (1 + ε)·p* for any constant ε > 0, then with probability 1 − o(1) the random graph can be edge‑colored with d colors so that it becomes rainbow‑k‑connected; conversely, if p = (1 − ε)·p*, then with probability 1 − o(1) rcₖ(G) > d. The proof proceeds in two parts.

For the upper‑bound direction, the authors first show that when p exceeds p* the graph almost surely has minimum degree at least d. They then construct, for each unordered vertex pair {u,v}, a collection of Θ(log n) candidate internally disjoint u‑v paths of length at most d. By assigning colors uniformly at random from a palette of d colors and applying a sophisticated “color‑matching” argument—essentially a multi‑matching version of Hall’s theorem—they prove that with high probability one can select k of those paths whose edges are all differently colored. The analysis uses Chernoff bounds to control the number of available paths, a union bound over all vertex pairs, and a Markov chain concentration argument to bound the probability of color collisions.

For the lower‑bound direction, the authors employ a first‑moment calculation to show that if p is below p* the graph almost surely contains vertices of degree at most d − 1. Such low‑degree vertices cannot be part of a d‑color rainbow‑k‑connected structure, forcing rcₖ(G) > d. This establishes the sharpness of p*.

Beyond the probabilistic existence result, the paper presents a constructive polynomial‑time algorithm that, with probability 1 − o(1), colors G(n,p) using at most one more color than the optimum. The algorithm works in two phases. In the first phase it extracts a collection of “core” subgraphs that exhibit the required d‑regular structure, using a BFS‑based search that runs in O(n log n) time. In the second phase it applies the color‑matching procedure to each core, assigning d colors while guaranteeing k internally disjoint rainbow paths for every vertex pair inside the core. Remaining edges are colored greedily, which never introduces new conflicts because the cores already provide the necessary connectivity. The overall running time is O(m log n) (m = number of edges), and the algorithm succeeds for a wide range of edge‑probabilities p = n^{−ε(1 ± o(1))} with any constant ε ∈