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/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008). doi:10.1007/978-1-84628-970-5

[2] Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) #57.

[3] S. Chakraborty, E. Fischer, A. Matsliah, and R. Yuster, Hardness and algorithms for rainbow connectivity, J. Comb. Optim. 21 (2011) 330-347. doi:10.1007/s10878-009-9250-9

[4] L.S. Chandran, A. Das, D. Rajendraprasad, and N.M. Varma, Rainbow connection number and connected dominating sets, J. Graph Theory 71 (2012) 206-218. doi:10.1002/jgt.20643

[5] G. Chartrand, G.L. Johns, K.A. McKeon, and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85-98.

[6] A.B. Ericksen, A matter of security, Graduating Engineer & Computer Careers (2007) 24-28.

[7] J. Ekstein, P. Holub, T. Kaiser, M. Koch, S. Matos Camacho, Z. Ryjáček and I. Schiermeyer, The rainbow connection number in 2-connected graphs, Discrete Math. doi:10.1016/j.disc.2012.04.022

[8] E. Győri, On division of graphs to connected subgraphs, Combinatorics, Vol. I, pp. 485-494, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam, 1978.

[9] A. Kemnitz and I. Schiermeyer, Graphs with rainbow connection number two, Discuss. Math. Graph Theory 31 (2011) 313-320. doi:10.7151/dmgt.1547

[10] M. Krivelevich and R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree, J. Graph Theory 63 (2010) 185-191.

[11] V.B. Le and Zs. Tuza, Finding optimal rainbow connection is hard, Preprint 2009.

[12] X. Li, M. Liu, and I. Schiermeyer, Rainbow connection number of dense graphs, to appear in Discuss. Math. Graph Theory. arXiv: 1110.5772v1 [math.CO] 2011

[13] X. Li and Y. Sun, Rainbow Connections of Graphs, Springer Briefs in Math., Springer, New York, 2012.

[14] L. Lovász, A homology theory for spanning trees of a graph, Acta Math. Hungar. 30 (1977) 241-251. doi:10.1007/BF01896190

[15] I. Schiermeyer, Rainbow connection in graphs with minimum degree three, Lecture Notes Computer Science 5874 (2009) 432-437. doi:10.1007/978-3-642-10217-2 42