## GRAPHS WITH RAINBOW CONNECTION NUMBER TWO

 Arnfried Kemnitz Computational Mathematics Technische Universität Braunschweig 38023 Braunschweig, Germany e-mail: a.kemnitz@tu-bs.de Ingo Schiermeyer Institut für Diskrete Mathematik und Algebra Technische Universität Bergakademie Freiberg 09596 Freiberg, Germany e-mail: Ingo.Schiermeyer@tu-freiberg.de

## Abstract

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−1) || 2)+1 ≤ m ≤ (n || 2) −1. We also characterize graphs with rainbow connection number two and large clique number.

Keywords: edge colouring, rainbow colouring, rainbow connection.

2010 Mathematics Subject Classification: 05C15, 05C35.

## References

 [1] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008), doi: 10.1007/978-1-84628-970-5. [2] S. Chakraborty, E. Fischer, A. Matsliah and R. Yuster, Hardness and algorithms for rainbow connectivity, Proceedings STACS 2009, to appear in Journal of Combinatorial Optimization. [3] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster On rainbow connection, Electronic J. Combin. 15 (2008) #57. [4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85-98. [5] A.B. Ericksen, A matter of security, Graduating Engineer & Computer Careers (2007) 24-28. [6] 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. [7] V.B. Le and Z. Tuza, Finding optimal rainbow connection is hard, preprint 2009. [8] I. Schiermeyer, Rainbow connection in graphs with minimum degree three, IWOCA 2009, LNCS 5874 (2009) 432-437.