Discussiones Mathematicae Graph Theory 33(1) (2013) 181-192
doi: 10.7151/dmgt.1640

Dedicated to Mietek Borowiecki on the occasion of his seventieth birthday

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Rainbow Connection in Sparse Graphs

Arnfried Kemnitz

Computational Mathematics, Technische Universität Braunschweig
38 023 Braunschweig, Germany

Jakub Przybyło and Mariusz Wożniak

AGH University of Science and Technology
al. A. Mickiewicza 30, 30-059 Kraków, Poland

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
Technische Universität Bergakademie Freiberg
09 596 Freiberg, Germany

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