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LIPIcs - Leibniz International Proceedings in Informatics. 26th International Symposium on Theoretical Aspects of Computer Science
Doi
Abstract
An edge-colored graph G is {em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {em 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 NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G)=2, and also that it is NP-Complete to decide whether a given edge-colored (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 epsilonn has bounded rainbow connectivity, where the bound depends only on epsilon, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented
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