Hardness and algorithms for rainbow connection

Sourav Chakraborty (), Eldar Fischer (), Arie Matsliah () and Raphael Yuster ()
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Sourav Chakraborty: University of Chicago
Eldar Fischer: Technion
Arie Matsliah: Centrum Wiskunde & Informatica (CWI)
Raphael Yuster: University of Haifa

Journal of Combinatorial Optimization, 2011, vol. 21, issue 3, No 4, 330-347

Abstract: 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.

Keywords: Connectivity; Rainbow coloring (search for similar items in EconPapers)
Date: 2011
References: View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10878-009-9250-9

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