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.

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,. . .,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but K 2 , is antimagic. Our main result validates this conjecture for graphs having minimum degree (log n). The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but K 2 ) and graphs with maximum degree at least n À 2 are antimagic.

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