# Graphs with rainbow connection number two

Arnfried Kemnitz; Ingo Schiermeyer

Discussiones Mathematicae Graph Theory (2011)

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

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topArnfried 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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- [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] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster On rainbow connection, Electronic J. Combin. 15 (2008) #57.
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- [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. Zbl1193.05079
- [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. Zbl1267.05125

## Citations in EuDML Documents

top- Xueliang Li, Mengmeng Liu, Ingo Schiermeyer, Rainbow Connection Number of Dense Graphs
- Arnfried Kemnitz, Jakub Przybyło, Ingo Schiermeyer, Mariusz Woźniak, Rainbow Connection In Sparse Graphs
- Ingo Schiermeyer, Bounds for the rainbow connection number of graphs
- Sandi Klavžar, Gašper Mekiš, On the rainbow connection of Cartesian products and their subgraphs

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