Geodesic
Graphs with rainbow connection number two
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 313-320

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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 prove that rc(G) = 2 for every connected graph G of order n and size m, where n-12 + 1 ≤ m ≤ n2 - 1. We also characterize graphs with rainbow connection number two and large clique number.
Keywords: edge colouring, rainbow colouring, rainbow connection 
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Kemnitz, Arnfried; Schiermeyer, Ingo. Graphs with rainbow connection number two. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 313-320. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a7/