#### Date of Original Version

10-7-2012

#### Type

Article

#### Rights Management

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-32512-0_46

#### Abstract or Description

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=logn+ωn where *ω* = *ω*(*n*) → ∞ and *ω* = *o*(log*n*) and of random *r*-regular graphs where *r* ≥ 3 is a fixed integer. Specifically, we prove that the rainbow connectivity *rc*(*G*) of *G* = *G*(*n*,*p*) satisfies rc(G)∼max{Z1,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 lognloglogn *whp*. Finally, we prove that the rainbow connectivity *rc*(*G*) of the random *r*-regular graph *G* = *G*(*n*,*r*) satisfies *rc*(*G*) = *O*(log2 *n*) *whp*.

#### DOI

10.1007/978-3-642-32512-0_46

#### Published In

Lecture Notes in Computer Science, 7408, 541-552.