Journal of Combinatorial Optimization

, Volume 21, Issue 3, pp 330–347 | Cite as

Hardness and algorithms for rainbow connection

  • Sourav Chakraborty
  • Eldar Fischer
  • Arie Matsliah
  • Raphael YusterEmail author


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 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 the first result of this paper we prove that computing rc(G) is NP-Hard solving an open problem from Caro et al. (Electron. J. Comb. 15, 2008, Paper R57). 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 ε>0, a connected graph with minimum degree at least ε n has bounded rainbow connection, where the bound depends only on ε, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.


Connectivity Rainbow coloring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alon N, Spencer JH (2000) The probabilistic method, 2nd edn. Wiley, New York CrossRefzbMATHGoogle Scholar
  2. Alon N, Duke RA, Lefmann H, Rödl V, Yuster R (1994) The algorithmic aspects of the Regularity Lemma. J Algorithms 16:80–109 CrossRefzbMATHMathSciNetGoogle Scholar
  3. Blum A, Karger D (1997) An \(\tilde{O}(n^{3/14})\) -coloring algorithm for 3-colorable graphs. Inf Process Lett 61(1):49–53 CrossRefMathSciNetGoogle Scholar
  4. Caro Y, Lev A, Roditty Y, Tuza Z, Yuster R (2008) On rainbow connection, Electron J Comb 15, Paper R57 Google Scholar
  5. Chartrand G, Johns GL, McKeon KA, Zhang P (2008) Rainbow connection in graphs. Math Bohem 133(1):85–98 zbMATHMathSciNetGoogle Scholar
  6. Fischer E, Matsliah A, Shapira A (2007) Approximate hypergraph partitioning and applications. In: Proceedings of the 48th annual IEEE symposium on foundations of computer science (FOCS), pp 579–589 Google Scholar
  7. Komlós J, Simonovits M (1996) Szemerédi’s Regularity Lemma and its applications in graph theory. In: Miklós, D, Sós, VT, Szönyi, T (eds) Combinatorics, Paul Erdös is Eighty. Bolyai society mathematical studies, vol 2. Budapest, pp 295–352 Google Scholar
  8. Szemerédi E (1978) Regular partitions of graphs. In: Proc. colloque inter. CNRS 260. CNRS, Paris, pp 399–401. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Sourav Chakraborty
    • 1
  • Eldar Fischer
    • 2
  • Arie Matsliah
    • 3
  • Raphael Yuster
    • 4
    Email author
  1. 1.Department of Computer ScienceUniversity of ChicagoChicagoUSA
  2. 2.Department of Computer ScienceTechnionHaifaIsrael
  3. 3.Centrum Wiskunde & Informatica (CWI)AmsterdamNetherlands
  4. 4.Department of MathematicsUniversity of HaifaHaifaIsrael

Personalised recommendations