Bounds for the rainbow connection number of graphs

Ingo Schiermeyer

Discussiones Mathematicae Graph Theory (2011)

  • Volume: 31, Issue: 2, page 387-395
  • ISSN: 2083-5892

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 show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.

How to cite

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Ingo Schiermeyer. "Bounds for the rainbow connection number of graphs." Discussiones Mathematicae Graph Theory 31.2 (2011): 387-395. <http://eudml.org/doc/270874>.

@article{IngoSchiermeyer2011,
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 show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.},
author = {Ingo Schiermeyer},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {rainbow colouring; rainbow connectivity; extremal problem; bounds; rainbow connection number},
language = {eng},
number = {2},
pages = {387-395},
title = {Bounds for the rainbow connection number of graphs},
url = {http://eudml.org/doc/270874},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Ingo Schiermeyer
TI - Bounds for the rainbow connection number of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 387
EP - 395
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 show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.
LA - eng
KW - rainbow colouring; rainbow connectivity; extremal problem; bounds; rainbow connection number
UR - http://eudml.org/doc/270874
ER -

References

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  1. [1] J.C. Bermond, On Hamiltonian Walks, Proc. of the Fifth British Combinatorial Conference, Aberdeen, 1975, Utlitas Math. XV (1976) 41-51. 
  2. [2] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008), doi: 10.1007/978-1-84628-970-5. 
  3. [3] S. Chakraborty, E. Fischer, A. Matsliah and R. Yuster, Hardness and algorithms for rainbow connectivity, Proceedings STACS 2009, to appear in J. Combin. Optim. Zbl1236.68080
  4. [4] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster, On rainbow connection, Electronic J. Combin. 15 (2008) #57. Zbl1181.05037
  5. [5] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85-98. Zbl1199.05106
  6. [6] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69. Zbl0047.17001
  7. [7] A.B. Ericksen, A matter of security, Graduating Engineer & Computer Careers (2007) 24-28. 
  8. [8] A. Kemnitz and I. Schiermeyer, Graphs with rainbow connection number two, Discuss. Math. Graph Theory 31 (2011) 313-320, doi: 10.7151/dmgt.1547. 
  9. [9] 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
  10. [10] V.B. Le and Z. Tuza, Finding optimal rainbow connection is hard, preprint 2009. 
  11. [11] I. Schiermeyer, Rainbow connection in graphs with minimum degree three, International Workshop on Combinatorial Algorithms, IWOCA 2009, LNCS5874 (2009) 432-437. Zbl1267.05125

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