PDF
Discussiones Mathematicae Graph Theory 31(2) (2011)
387-395
DOI: https://doi.org/10.7151/dmgt.1553
Bounds for the Rainbow Connection Number of Graphs
| Ingo Schiermeyer
Institut für Diskrete Mathematik und Algebra |
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.Keywords: rainbow colouring, rainbow connectivity, extremal problem
2010 Mathematics Subject Classification: 05C35, 05C15.
References
| [1] | J.C. Bermond, On Hamiltonian Walks, Proc. of the Fifth British Combinatorial Conference, Aberdeen, 1975, Utlitas Math. XV (1976) 41--51. |
| [2] | J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008), doi: 10.1007/978-1-84628-970-5. |
| [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. |
| [4] | Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster, On rainbow connection, Electronic J. Combin. 15 (2008) #57. |
| [5] | G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85--98. |
| [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. |
| [7] | A.B. Ericksen, A matter of security, Graduating Engineer & Computer Careers (2007) 24--28. |
| [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] | 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. |
| [10] | V.B. Le and Z. Tuza, Finding optimal rainbow connection is hard, preprint 2009. |
Received 5 January 2010
Revised 14 January 2011
Accepted 17 January 2011
Close