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Hardness and Algorithms for Rainbow Connectivity

Abstract : An edge-colored 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 t he 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 ǫ > 0, a connected graph with minimum degree at least ǫn has bounded rainbow connectivity, where the bound depends only on ǫ, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also pre sented.
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https://hal.inria.fr/inria-00359276
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Submitted on : Monday, February 16, 2009 - 3:44:31 PM
Last modification on : Wednesday, December 20, 2017 - 5:42:07 PM
Long-term archiving on: : Wednesday, September 22, 2010 - 11:34:38 AM

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  • HAL Id : inria-00359276, version 2
  • ARXIV : 0902.1255

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Sourav Chakraborty, Eldar Fischer, Arie Matsliah, Raphael Yuster. Hardness and Algorithms for Rainbow Connectivity. 26th International Symposium on Theoretical Aspects of Computer Science STACS 2009, Feb 2009, Freiburg, Germany. pp.243-254. ⟨inria-00359276v2⟩

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