Geodesic
Rainbow Connection In Sparse Graphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 181-192

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An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and |E(G)| ≥n-k+12 + k -1 for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.
Keywords: rainbow-connected graph, rainbow colouring, rainbow connection number 
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Kemnitz, Arnfried; Przybyło, Jakub; Schiermeyer, Ingo; Woźniak, Mariusz. Rainbow Connection In Sparse Graphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 181-192. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a14/