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
The (strong) rainbow connection numbers of Cayley graphs on Abelian groups
Computers & Mathematics with Applications
On rainbow-k-connectivity of random graphs
Information Processing Letters
Rainbow connection number and connected dominating sets
Journal of Graph Theory
A survey of Nordhaus-Gaddum type relations
Discrete Applied Mathematics
On minimally rainbow k-connected graphs
Discrete Applied Mathematics
Rainbow connection and minimum degree
Discrete Applied Mathematics
Rainbow vertex k-connection in graphs
Discrete Applied Mathematics
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An edge-colored graph Gis rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make Grainbow edge-connected. We prove that if Ghas nvertices and minimum degree δ then rc(G)20n-δ. This solves open problems from Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243–254). A vertex-colored graph Gis rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make Grainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if Ghas nvertices and minimum degree δ then rvc(G)11n-δ. We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 185–191, 2010