# Graphs with rainbow connection number two

• Volume: 31, Issue: 2, page 313-320
• ISSN: 2083-5892

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## Abstract

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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 $\left(\genfrac{}{}{0pt}{}{n-1}{2}\right)+1\le m\le \left(\genfrac{}{}{0pt}{}{n}{2}\right)-1$. We also characterize graphs with rainbow connection number two and large clique number.

## How to cite

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Arnfried Kemnitz, and Ingo Schiermeyer. "Graphs with rainbow connection number two." Discussiones Mathematicae Graph Theory 31.2 (2011): 313-320. <http://eudml.org/doc/271041>.

@article{ArnfriedKemnitz2011,
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 $\binom\{n-1\}\{2\} + 1 ≤ m ≤ \binom\{n\}\{2\} - 1$. We also characterize graphs with rainbow connection number two and large clique number.},
author = {Arnfried Kemnitz, Ingo Schiermeyer},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {edge colouring; rainbow colouring; rainbow connection},
language = {eng},
number = {2},
pages = {313-320},
title = {Graphs with rainbow connection number two},
url = {http://eudml.org/doc/271041},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Arnfried Kemnitz
AU - Ingo Schiermeyer
TI - Graphs with rainbow connection number two
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 313
EP - 320
AB - 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 $\binom{n-1}{2} + 1 ≤ m ≤ \binom{n}{2} - 1$. We also characterize graphs with rainbow connection number two and large clique number.
LA - eng
KW - edge colouring; rainbow colouring; rainbow connection
UR - http://eudml.org/doc/271041
ER -

## References

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1. [1] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008), doi: 10.1007/978-1-84628-970-5.
2. [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. Zbl1236.68080
3. [3] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster On rainbow connection, Electronic J. Combin. 15 (2008) #57.
4. [4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85-98. Zbl1199.05106
5. [5] A.B. Ericksen, A matter of security, Graduating Engineer & Computer Careers (2007) 24-28.
6. [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. Zbl1193.05079
7. [7] V.B. Le and Z. Tuza, Finding optimal rainbow connection is hard, preprint 2009.
8. [8] I. Schiermeyer, Rainbow connection in graphs with minimum degree three, IWOCA 2009, LNCS 5874 (2009) 432-437. Zbl1267.05125

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