Hardness and algorithms for rainbow connection |
| |
Authors: | Sourav Chakraborty Eldar Fischer Arie Matsliah Raphael Yuster |
| |
Affiliation: | 1.Department of Computer Science,University of Chicago,Chicago,USA;2.Department of Computer Science,Technion,Haifa,Israel;3.Centrum Wiskunde & Informatica (CWI),Amsterdam,Netherlands;4.Department of Mathematics,University of Haifa,Haifa,Israel |
| |
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: | |
本文献已被 SpringerLink 等数据库收录! |
|