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.

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A vertex set Y in a (hyperbraph is called k-independent if in the sub(hyper)-graph induced by Y every vertex is incident to less than k edges. We prove a lower bound for the maximum cardinality of a k-independent set-in terms of degree sequences-which strengthens and generalizes several previously known results, including Turin's theorem.

Given a simple undirected graph G and a positive integer k, the k-forcing number of G, denoted F k (G), is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most k non-colored neighbors, then each of its non-colored neighbors becomes colored. When k = 1, this is equivalent to the zero forcing number, usually denoted with Z(G), a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the k-forcing number. Notable among these, we show that if G is a graph with order n ≥ 2 and maximum degree ∆ ≥ k, then F k (G) ≤ (∆−k+1)n ∆−k+1+min {δ,k} . This simplifies to, for the zero forcing number case of k = 1, Z(G) = F 1 (G) ≤ ∆n ∆+1 . Moreover, when ∆ ≥ 2 and the graph is k-connected, we prove that F k (G) ≤ (∆−2)n+2 ∆+k−2 , which is an improvement when k ≤ 2, and specializes to, for the zero forcing number case, Z(G) = F 1 (G) ≤ (∆−2)n+2 ∆−1 . These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the k-forcing number and the connected k-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.