Discussiones Mathematicae Graph Theory 31(2) (2011) 313-320
doi: 10.7151/dmgt.1547

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GRAPHS WITH RAINBOW CONNECTION NUMBER TWO

Arnfried Kemnitz

Computational Mathematics
Technische Universität Braunschweig
38023 Braunschweig, Germany
e-mail: a.kemnitz@tu-bs.de

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
Technische Universität Bergakademie Freiberg
09596 Freiberg, Germany
e-mail: Ingo.Schiermeyer@tu-freiberg.de

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 ((n−1) || 2)+1 ≤ m ≤ (n || 2) −1. We also characterize graphs with rainbow connection number two and large clique number.

Keywords: edge colouring, rainbow colouring, rainbow connection.

2010 Mathematics Subject Classification: 05C15, 05C35.

References

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Received 4 December 2009
Revised 12 May 2010
Accepted 12 May 2010