Rainbow Connectivity of Sparse Random Graphs

TL;DR

This paper investigates the rainbow connectivity of sparse random graphs and regular graphs, establishing asymptotic bounds and exact values for the minimum number of colors needed for rainbow connectivity.

Contribution

It provides the first detailed analysis of rainbow connectivity at the connectivity threshold for binomial and regular random graphs, including asymptotic formulas and bounds.

Findings

Rainbow connectivity of G(n,p) is asymptotically max{Z_1, diameter(G)}.

Rainbow connectivity of random r-regular graphs is O(log^2 n).

Results hold with high probability as n grows large.

Abstract

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\om}{n}$ where $\om=\om(n)\to\infty$ and ${\om}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\set{Z_1,diameter(G)}$ with high probability (\whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\diam$ \whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ satisfies $rc(G) =O(\log^2{n})$ \whp.