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.

We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem for two forbidden subgraphs.

Let F = {F1,…} be a given class of forbidden graphs. A graph G is called F‐saturated if no Fi ∈ F is a subgraph of G but the addition of an arbitrary new edge gives a forbidden subgraph. In this paper the minimal number of edges in F‐saturated graphs is examined. General estimations are given and the structure of minimal graphs is described for some special forbidden graphs (stars, paths, m pairwise disjoint edges).

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