11.00 a.m. Hardness and Algorithms for Rainbow Connectivity Sourav Chakraborty University of Chicago, U.S.A. 230708 Abstract An edgecolored graph G is "rainbow connected" if any two vertices are connected by a path whose edges have distinct colors. The "rainbow connectivity" 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 addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NPHard. In fact, we prove that it is already NPComplete to decide if rc(G)=2, and also that it is NPComplete to decide whether a given edgecolored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every \epsilon >0, a connected graph with minimum degree at least \epsilon n has bounded rainbow connectivity, where the bound depends only on \epsilon, and the corresponding coloring can be constructed in polynomial time. Additional nontrivial upper bounds, as well as open problems and conjectures are also presented
