### On Rainbow Connection

#### Abstract

An edge-colored graph $G$ is *rainbow connected* if any two vertices are connected by a path whose edges have distinct colors. The *rainbow connection number* of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we prove several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee $rc(G)=2$. Among our results we prove that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc(G) < 5n/6$, and if the minimum degree is $\delta$ then $rc(G) \le {\ln \delta\over\delta}n(1+o_\delta(1))$. We also determine the threshold function for a random graph to have $rc(G)=2$ and make several conjectures concerning the computational complexity of rainbow connection.