The algorithmic aspects of the regularity lemma
Journal of Algorithms
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
Approximate Hypergraph Partitioning and Applications
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
The complexity of determining the rainbow vertex-connection of a graph
Theoretical Computer Science
On the rainbow connectivity of graphs: complexity and FPT algorithms
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Rainbow connection number and connected dominating sets
Journal of Graph Theory
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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 驴0, a connected graph with minimum degree at least 驴 n has bounded rainbow connection, where the bound depends only on 驴, 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.