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Discussiones Mathematicae Graph Theory 33(1) (2013)
181-192
DOI: https://doi.org/10.7151/dmgt.1640
Dedicated to Mietek Borowiecki on the occasion of his seventieth birthday
Rainbow Connection in Sparse Graphs
| Arnfried Kemnitz
Computational Mathematics, Technische Universität Braunschweig |
Jakub Przybyło and Mariusz Wożniak
AGH University of Science and Technology | Ingo Schiermeyer
Institut für Diskrete Mathematik und Algebra |
Abstract
An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, G is rainbow-connected. In this paper we prove that denoted by rc(G), is the minimum number of colours such that rc(G) ≤ k if |V(G) | = n and |E(G) | ≥ ((n −k+1) || 2) + k −1 for all integers n and k with n −6 ≤ k ≤ n −3. We also show that this bound is tight.
Keywords: rainbow-connected graph, rainbow colouring, rainbow connection number
2010 Mathematics Subject Classification: 05C15.
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Received 11 April 2012
Revised 11 July 2012
Accepted 11 July 2012
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