Geodesic
Bounds for the rainbow connection number of graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 387-395

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An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow-connected. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.
Keywords: rainbow colouring, rainbow connectivity, extremal problem 
Schiermeyer, Ingo. Bounds for the rainbow connection number of graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 387-395. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a13/
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