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Hardness and algorithms for rainbow connection

Author

Listed:
  • Sourav Chakraborty

    (University of Chicago)

  • Eldar Fischer

    (Technion)

  • Arie Matsliah

    (Centrum Wiskunde & Informatica (CWI))

  • Raphael Yuster

    (University of Haifa)

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 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.

Suggested Citation

  • Sourav Chakraborty & Eldar Fischer & Arie Matsliah & Raphael Yuster, 2011. "Hardness and algorithms for rainbow connection," Journal of Combinatorial Optimization, Springer, vol. 21(3), pages 330-347, April.
    • Handle: RePEc:spr:jcomop:v:21:y:2011:i:3:d:10.1007_s10878-009-9250-9
    DOI: 10.1007/s10878-009-9250-9
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    Cited by:

    1. Fu-Hsing Wang & Cheng-Ju Hsu, 2024. "Rainbow Connection Numbers of WK-Recursive Networks and WK-Recursive Pyramids," Mathematics, MDPI, vol. 12(7), pages 1-11, March.
    2. Qingqiong Cai & Xueliang Li & Yan Zhao, 2016. "The 3-rainbow index and connected dominating sets," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1142-1159, April.
    3. Yingbin Ma & Zaiping Lu, 2017. "Rainbow connection numbers of Cayley graphs," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 182-193, July.

    More about this item

    Keywords

    Connectivity; Rainbow coloring;

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