Hardness and algorithms for rainbow connectivity

Sourav Chakraborty, Eldar Fischer, Arie Matsliah, Raphael Yuster

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 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 ∈ > 0, a connected graph with minimum degree at least ∈n has bounded rainbow connectivity, where the bound depends only on ∈, 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.

Original languageEnglish
Title of host publicationSTACS 2009 - 26th International Symposium on Theoretical Aspects of Computer Science
Number of pages12
StatePublished - 2009
Event26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009 - Freiburg, Germany
Duration: 26 Feb 200928 Feb 2009

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009

ASJC Scopus subject areas

  • Software


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